Answers (Anticipation Guide and Lesson 71)


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1 Answers (Anticiption Guide nd Lesson 7) NAME DATE PERID 7 Anticiption Guide Rdicl Equtions STEP Chpter 7 Glencoe Algebr Answers Chpter Resources Before ou begin Chpter 7 Red ech sttement. Decide whether ou Agree (A) or Disgree (D) with the sttement. Write A or D in the first column R if ou re not sure whether ou gree or disgree, write NS (Not Sure). STEP Sttement STEP A, D, or NS A or D. Functions cn be dded or subtrcted in the sme w s polnomils. A. A composition of functions, f [g()], is found b multipling f() b g(). D. The inverse of function is the set of ordered pirs obtined b tking the opposite of ech coordinte in the originl D ordered pirs.. Two functions re inverses of ech other onl if their compositions re the identit function. A. The domin of ( ) would be. A. The principl root of n nth root is lws positive. D m m 7. The rdicl epression is in simplest form. D 8. nd re conjugtes of ech other. A 9. is the sme s. D 0. To solve n eqution contining the squre root of the vrible, squre both sides of the eqution. A STEP After ou complete Chpter 7 Rered ech sttement nd complete the lst column b entering n A or D. Did n of our opinions bout the sttements chnge from the first column? For those sttements tht ou mrk with D, use piece of pper to write n emple of wh ou disgree. Lesson 77 NAME DATE PERID Lesson Reding Guide pertions on Functions Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. Describe two ws to clculte Ms. Coffmon s profit from the sle of 0 birdhouses. (Do not ctull clculte her profit.) Smple nswer:. Find the revenue b substituting 0 for in the epression. Net, find the cost b substituting 0 for in the epression 00. Finll, subtrct the cost from the revenue to find the profit.. Form the profit function p() r() c() ( 00) Substitute 0 for in the epression Red the Lesson. Determine whether ech sttement is true or flse. (Remember tht true mens lws true.). If f nd g re polnomil functions, then f g is polnomil function. true f b. If f nd g re polnomil functions, then g is polnomil function. flse c. If f nd g re polnomil functions, the domin of the function f g is the set of ll rel numbers. true d. If f() nd g(), the domin of the function is the set of ll rel numbers. flse f g e. If f nd g re polnomil functions, then (f g)() (g f)(). flse f. If f nd g re polnomil functions, then (f g)() (g f)(). true. Let f() nd g().. Eplin in words how ou would find (f g)(). (Do not ctull do n clcultions.) Smple nswer: Squre nd dd. Tke the number ou get, multipl it b, nd subtrct. b. Eplin in words how ou would find (g f)(). (Do not ctull do n clcultions.) Smple nswer: Multipl b nd subtrct. Tke the number ou get, squre it, nd dd. Remember Wht You Lerned. Some students hve trouble remembering the correct order in which to ppl the two originl functions when evluting composite function. Write three sentences, ech of which eplins how to do this in slightl different w. (Hint: Use the word closest in the first sentence, the words inside nd outside in the second, nd the words left nd right in the third.) Smple nswer:. The function tht is written closest to the vrible is pplied first.. Work from the inside to the outside.. Work from right to left. Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
2 Answers (Lesson 7) Lesson 7 NAME DATE PERID NAME DATE PERID 7 Stud Guide nd Intervention (continued) pertions on Functions 7 Stud Guide nd Intervention Composition of Functions pertions on Functions Arithmetic pertions Composition Suppose f nd g re functions such tht the rnge of g is subset of the domin of f. of Functions Then the composite function f g cn be described b the eqution [f g]() f [g ()]. Sum (f g)() f() g() Difference (f g)() f() g() pertions with Functions Product (f g)() f() g() f f() g g() Emple Quotient (), g() 0 For f {(, ), (, ), (, ), (, )} nd g {(, ), (, ), (, ), (, )}, find f g nd g f if the eist. f[ g()] f() f[ g()] f() f[ g()] f() f[ g()] f() f g {(, ), (, ), (, ), (, )} g[f()] g() g[f()] g() g[f()] g() g[f()] g() g f {(, ), (, ), (, ), (, )} Emple Find [g h]() nd [h g]() for g() nd h(). [g h]() g[h()] [h g]() h[ g()] g( ) h( ) ( ) ( ) Eercises Emple f Find (f g)(), (f g)(), (f g)(), nd g () for f() nd g(). (f g)() f() g() Addition of functions ( ) ( ) f(), g() Simplif. (f g)() f() g() Subtrction of functions ( ) ( ) f(), g() Simplif. (f g)() f() g() Multipliction of functions ( )( ) f(), g() ( ) ( ) ( ) Distributive Propert 9 8 Distributive Propert Simplif. f () g f() Division of functions g() For ech set of ordered pirs, find f g nd g f if the eist., f() nd g(). f {(, ), (, ), (0, 9)},. f {(, ), (9, 8), (, ), (0, )}, g {(, 0), (, ), (9, )} g {(, 7), (, ), (, ), (8, 0)} f g {(, ), (, 9), (9, )}; f g does not eist; g f {(, ), (0, ), (, 0)} g f {(, 7), (0, ), (, ), (9, 0)} Eercises f Find (f g)(), (f g)(), (f g)(), nd g () for ech f() nd g().. f() 8 ; g(). f() ; g() ; 8; 8 ; 8; ; 8, 8 ;, Find [f g]() nd [g f]().. f() 7; g(). f() ; g() [f g]() 0, [f g](), [g f ]() 0 [g f ]() 8. f() ; g(). f() ; g() ; 8; ; 9 ; ; 9 9 ;,,, ( )( ). f() ; g() 9. f() ; g() [f g](), [f g]() 9, [g f ]() 9 [g f ](). f() ; g(), ;, ;, ;, Chpter 7 7 Glencoe Algebr Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
3 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Skills Prctice pertions on Functions f Find (f g)(), (f g)(), (f g)(), nd g () for ech f() nd g().. f() ; 9;. f() ; ; 7 ; g() 0; g(),,. f() ; ;. f(), 0; g() ;, g(), 0;, 0, 0; For ech set of ordered pirs, find f g nd g f if the eist.. f {(0, 0), (, )}. f {(0, ), (, ), (, )} g {(0, ), (, 0), (, 0)} g {(, ), (, 0)} {(0, ), (, 0), (, 0)}; {(, ), (, )}; {(0, ), (, 0)} {(0, ), (, 0), (, 0)} 7. f {(, ), (, ), (, )} 8. f {(, ), (, ), (, )} g {(, ), (, ), (, )} g {(, ), (, ), (, )} {(, ), (, ), (, )}; {(, ), (, ), (, )}; {(, ), (, ), (, )} {(, ), (, ), (, )} Find [g h]() nd [h g](). 9. g() ; 0. g() ; h() h(). g() ;. g() ; 9 h() h(). g() ;. g() ; 8 h() h() If f(), g(), nd h(), find ech vlue.. f[ g()]. g[h(0)] 7. g[f()] 8. h[f()] 9. g[h()] 0. h[f(0)] 899. f[h(8)] 89. [f (h g)]() 7. [f ( g h)]() Chpter 7 8 Glencoe Algebr Answers NAME DATE PERID 7 Prctice pertions on Functions Find (f g)(), (f g)(), (f g)(), nd g f () for ech f() nd g().. f(). f() 8. f() 7 g() g() g() 9 ; ; 8, 0; 7 ; 7 ; ; 8, 0; 7 08;, 8, 0; 8, 0 For ech set of ordered pirs, find f g nd g f if the eist.,. f {(9, ), (, 0), (, )}. f {(, ), (0, ), (, )} g {(0, 9), (, ), (, )} g {(, 0), (, )} {(0, ), (, ), (, 0)}; {(, ), (, )}; {(9, ), (, 9), (, )} {(, ), (0, 0), (, 0)}. f {(, ), (0, ), (, )} 7. f {(0, ), (, ), (, 8)} g {(, ), (, 0), (, )} g {(8, ), (, 0), (, )} {(, ), (, ), (, )}; does not eist; {(, 0), (0, ), (, )} {(0, 0), (, 0), (, )} Find [g h]() nd [h g](). 8. g() 9. g() 8 0. g() h() h() h() ; ; ; 08. g(). g(). g() h() h() h() ; ; ; 8 If f(), g(), nd h(), find ech vlue.. f[ g()]. g[h()] 0. h[f()] 0 7. f[h(9)] 8. h[ g()] 9. g[f(8)] 0 0. h[f(0)] 0. [f (h g)](). [f (g h)]() 00. BUSINESS The function f() models the mnufcturing cost per item when items re produced, nd g() models the service cost per item. Write function C() for the totl mnufcturing nd service cost per item. C() n. MEASUREMENT The formul f converts inches n to feet f, nd m feet to miles m. Write composition of functions tht converts inches to miles. n [m f ]n, 0 f converts 80 Chpter 7 9 Glencoe Algebr Chpter 7 A Glencoe Algebr
4 Answers (Lesson 7). AREA Bernrd wnts to know the re of figure mde b joining n equilterl tringle nd squre long n edge. The function f(s) s gives the re of n equilterl tringle with side s. The function g(s) s gives the re of squre with side s. Wht function h(s) gives the re of the figure s function of its side length s? h(s) (f g)(s) ( )s. PRICING A computer compn decides to continuousl djust the pricing of nd discounts to its products in n effort to remin competitive. The function P(t) gives the sle price of its Super000 computer s function of time. The function D(t) gives the vlue of specil discount it offers to vlued customers. How much would vlued customers hve to p for one Super000 computer? (P D)(t). LAVA A freshl ejected lv rock immeditel begins to cool down. The temperture of the lv rock in degrees Fhrenheit s function of time is given b T(t). Let C(F) be the function tht gives degrees Celsius s function of degrees Fhrenheit. Wht function gives the temperture of the lv rock in degrees Celsius s function of time? C(T(t)). ENGINEERING A group of engineers is designing stple gun. ne tem determines tht the speed of impct s of the stple (in feet per second) s function of the hndle length (in inches) is given b s() 0. A second tem determines tht the number of sheets N tht cn be stpled s function of the impct speed is given b N(s) s gives N s function of? N(s()) 0 0. Wht function HT AIR BALLNS For Eercises nd, use the following informtion. Hnnh nd Terr went on onehour hot ir blloon ride. Let T(A) be the outside ir temperture s function of ltitude nd let A(t) be the ltitude of the blloon s function of time Wht function describes the ir temperture Hnnh nd Terr felt t different times during their trip? T(A(t)). Sketch grph of the function ou wrote for Eercise bsed on the grphs for T(A) nd A(t) tht re given Teperture (ºF) Teperture (ºF) Altitude (km) Lesson 77 NAME DATE PERID Word Problem Prctice pertions on Functions s 0 Altitude (km) Time (minutes) Time (minutes) Chpter 7 0 Glencoe Algebr 7 NAME DATE PERID Enrichment Reltive Mimum Vlues The grph of f () 9 shows reltive mimum vlue somewhere between f () nd f (). You cn obtin closer pproimtion b compring vlues such s those shown in the tble. To the nerest tenth reltive mimum vlue for f () is.. f() 8 f() 9 f() Using clcultor to find points, grph ech function. To the nerest tenth, find reltive mimum vlue of the function.. f () ( ) rel. m. of.0. f () rel. m. of.0 f() f(). f () 9 rel. m. of 8.. f () rel. m. of. f() f() Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
5 Answers (Lessons 7 nd 7) Lesson 77 NAME DATE PERID Spredsheet Activit pertions on Functions It is possible to perform opertions on functions such s ddition, subtrction, multipliction nd division. You cn use spredsheet to investigte the reltionships mong functions. Consider the functions f(), g(), nd h() +. Find the function vlues of ech function for severl vlues of. Does it pper tht f() g() h()? Use Column A for the chosen vlues of. Columns B, C, nd E re f(), g(), nd h() respectivel. Use Column D for f() g(). For ever vlue of, f() g() h(). Functions.ls A. 0 B f() 0. 8 C g() D f() + g().7 Sheet Sheet Sheet 8 E h().7 8 Eercises Stud nd use the spredsheet bove.. Find k() ( ) ( ). How does it compre to h()? k() h(). Chnge the functions in the spredsheet to f(), g(), nd h(). How re these functions relted? Is it true tht f() g() h()? (f g)() h(); es. Mke conjecture bout (f g)() for n functions f() nd g(). (f g)() f() g(). Mke conjecture bout (f g)() for n functions f() nd g(). Use the spredsheet to test our conjecture. Does it pper to be true? Eplin our nswer. (f g)() f() g(); See students work. Find (f g)(), (f g)(), for ech f() nd g(). Use the spredsheet to find function vlues to verif our solutions. 7. See students spredsheets.. f() 8. f() 7. f() 0 g() 9 g() g() 7 7; ; 9 ; Chpter 7 Glencoe Algebr Answers 7 NAME DATE PERID Lesson Reding Guide Inverse Functions nd Reltions Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. A function multiplies number b nd then dds to the result. Wht does the inverse function do, nd in wht order? Smple nswer: It first subtrcts from the number nd then divides the result b. Red the Lesson. Complete ech sttement.. If two reltions re inverses, the domin of one reltion is the rnge of the other. b. Suppose tht g is reltion nd tht the point (, ) is on its grph. Then point on the grph of g is (, ). c. The horizontl line test cn be used on the grph of function to determine whether the function hs n inverse function. d. If ou re given the grph of function, ou cn find the grph of its inverse b reflecting the originl grph over the line with eqution. e. If f nd g re inverse functions, then (f g)() nd (g f)(). f. A function hs n inverse tht is lso function onl if the given function is onetoone g. Suppose tht h() is function whose inverse is lso function. If h(), then h ().. Assume tht f() is onetoone function defined b n lgebric eqution. Write the four steps ou would follow in order to find the eqution for f ().. Replce f() with in the originl eqution.. Interchnge nd.. Solve for.. Replce with f (). Remember Wht You Lerned. A good w to remember something new is to relte it to something ou lred know. How re the verticl nd horizontl line tests relted? Smple nswer: The verticl line test determines whether reltion is function becuse the ordered pirs in function cn hve no repeted vlues. The horizontl line test determines whether function is onetoone becuse onetoone function cnnot hve n repeted vlues. Chpter 7 Glencoe Algebr. Chpter 7 A Glencoe Algebr
6 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Stud Guide nd Intervention Inverse Functions nd Reltions Find Inverses Inverse Reltions Two reltions re inverse reltions if nd onl if whenever one reltion contins the element (, b), the other reltion contins the element (b, ). Propert of Inverse Suppose f nd f re inverse functions. Functions Then f() b if nd onl if f (b). Emple Find the inverse of the function f(). Then grph the function nd its inverse. Step Replce f() with in the originl eqution. f() f() Step Interchnge nd. Step Solve for. Inverse Multipl ech side b. Add to ech side. ( ) Divide ech side b. f() f () The inverse of f() is f () ( ). Eercises Find the inverse of ech function. Then grph the function nd its inverse.. f(). f(). f() f () f () f () 8 f() f() f() f () f () f() f() f() f () 8 Chpter 7 Glencoe Algebr NAME DATE PERID 7 Stud Guide nd Intervention (continued) Inverse Functions nd Reltions Inverses of Reltions nd Functions Inverse Functions Two functions f nd g re inverse functions if nd onl if [f g]() nd [g f ](). Emple Determine whether f() 7 nd g() ( 7) re inverse functions. [ f g]() f[ g()] [ g f ]() g[ f()] f ( 7) g( 7) ( 7) 7 ( 7 7) 7 7 The functions re inverses since both [ f g]() nd [ g f ](). Emple Determine whether f() f nd g() re inverse functions. [ f g]() f[ g()] Since [ f g](), the functions re not inverses. Eercises Determine whether ech pir of functions re inverse functions.. f(). f(). f() 0 g() es g() 0 es g() 0 no. f(). f() 8. f() g() no g() 8 no g() es 7. f() 8. f() 9. f() g() 8 es g() 0 ( ) es g() no 0. f() 0. f(). f() 9 g() 0 es g() es g() es Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
7 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Skills Prctice Inverse Functions nd Reltions Find the inverse of ech reltion.. {(, ), (, ), (8, )}. {(7, ), (0, ), (, )} {(, ), (, ), (, 8)} {(, 7), (, 0), (, )}. {(0, ), (7, ), (, ), (, 0)}. {(0, 9), (, ), (, ), (8, )} {(, 0), (, 7), (, ), (0, )} {(9, 0), (, ), (, ), (, 8)}. {(, ), (0, 7), (9, ), (0, )}. {(, ), (, ), (0, 8), (8, 9)} {(, ), (7, 0), (, 9), (, 0)} {(, ), (, ), (8, 0), (9, 8)} Find the inverse of ech function. Then grph the function nd its inverse f() 9. f() f () f () f() f() 0. g(). h(). g () g() h () h() Determine whether ech pir of functions re inverse functions.. f() no. f() es. f() es g() g() ( ) g(). f() es 7. h() no 8. f() 8 0 es g() g() g() 8 Chpter 7 Glencoe Algebr Answers NAME DATE PERID 7 Prctice Inverse Functions nd Reltions Find the inverse of ech reltion.. {(0, ), (, ), (, )}. {(, ), (, ), (, 8)} {(, 0), (, ), (, )} {(, ), (, ), (8, )}. {(, 7), (0, ), (, 9), (7, )}. {(8, ), (0, ), (, ), (, 7)} {(7, ), (, 0), (9, ), (, 7)} {(, 8), (, 0), (, ), (7, )}. {(, ), (, ), (, ), (7, 8)}. {(, 9), (, ), (0, 0), (, )} {(, ), (, ), (, ), (8, 7)} {(9, ), (, ), (0, 0), (, )} Find the inverse of ech function. Then grph the function nd its inverse. f () g () 7. f() 8. g() 9. f() g() Determine whether ech pir of functions re inverse functions. 0. f() es. f() es. g() no g() g() ( ) h(). f() no. f() es. g() 8 es 7 g() g() 7 h(). MEASUREMENT The points (, ), (7, 80), (7, 0), (, 08), nd (7, ) give the weight in pounds s function of height in inches for students in clss. Give the points for these students tht represent height s function of weight. (, ), (80, 7), (0, 7), (08, ), (, 7) REMDELING For Eercises 7 nd 8, use the following informtion. The Clers re replcing the flooring in their foot b 8 foot kitchen. The new flooring costs $7.99 per squre rd. The formul f() 9 converts squre rds to squre feet. 7. Find the inverse f (). Wht is the significnce of f () for the Clers? f () 9 ; It will llow them to convert the squre footge of their kitchen floor to squre rds, so the cn then clculte the cost of the new flooring. 8. Wht will the new flooring cost the Cler s? $9.70 Chpter 7 7 Glencoe Algebr Chpter 7 A7 Glencoe Algebr
8 Answers (Lesson 7) Lesson VLUME Json wnts to mke sphericl wter cooler tht cn hold hlf cubic meter of wter. He knows tht V r, but he needs to know how to find r given V. Find this inverse function. r NAME DATE PERID Word Problem Prctice Inverse Functions nd Reltions V. SELFINVERTIBLE Kren finds the incomplete grph of function in the bck of her engineering hndbook. The function is grphed in the figure below. Kren knows tht this function is its own inverse. Armed with this knowledge, etend the grph for vlues of between 7 nd. . EXERCISE Ale begn new eercise routine. To gin the mimum benefit from his eercise, Ale clculted his mimum trget hert rte using the function. f() 0.8(0 ) where represents his ge. Find the inverse of this function. f () PLANETS For Eercises nd, use the following informtion. The pproimte distnce of plnet from the Sun is given b d T where d is distnce in stronomicl units nd T is Erth ers. An stronomicl unit is the distnce of the Erth from the Sun. . Solve for T in terms of d. T d.. RCKETS The ltitude of rocket in feet s function of time is given b f(t) = 9t, where t 0. Find the inverse of this function nd determine the times when the rocket will be 0, 00, nd 000 feet high. Round our nswers to the nerest hundredth of second. t 7 f (t) ; 0 ft t 0. s, 00 ft t. s, 000 ft t. s. Pluto is bout 9. times s fr from the Sun s the Erth. About how mn ers does it tke Pluto to orbit the Sun? 8 r Chpter 7 8 Glencoe Algebr 7 NAME DATE PERID Enrichment Reding Algebr In mthemtics, the term group hs specil mening. The following numbered sentences discuss the ide of group nd one interesting emple of group. 0 To be group, set of elements nd binr opertion must stisf four conditions: the set must be closed under the opertion, the opertion must be ssocitive, there must be n identit element, nd ever element must hve n inverse. 0 The following si functions form group under the opertion of composition of functions: f (), f (), f (), f () ( ), f (), nd f ( ) (). ( ) 0 This group is n emple of noncommuttive group. For emple, f f f, but f f f. 0 Some eperimenttion with this group will show tht the identit element is f. 0 Ever element is its own inverse ecept for f nd f, ech of which is the inverse of the other. Use the prgrph to nswer these questions.. Eplin wht it mens to s tht set is closed under n opertion. Is the set of positive integers closed under subtrction? Performing the opertion on n two elements of the set results in n element of the sme set. No, nd re positive integers but is not.. Subtrction is noncommuttive opertion for the set of integers. Write n informl definition of noncommuttive. The order in which the elements re used with the opertion cn ffect the result.. For the set of integers, wht is the identit element for the opertion of multipliction? Justif our nswer., becuse, for ever integer, nd.. Eplin how the following sttement reltes to sentence 0: (f f )() f [ f ()] f ( f ) (). ( ) It shows tht f is the inverse of f. Chpter 7 9 Glencoe Algebr Chpter 7 A8 Glencoe Algebr
9 Answers (Lesson 7) Lesson 77 NAME DATE PERID Lesson Reding Guide Squre Root Functions Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. If the weight to be supported b steel cble is doubled, should the dimeter of the support cble lso be doubled? If not, b wht number should the dimeter be multiplied? no; Red the Lesson. Mtch ech squre root function from the list on the left with its domin nd rnge from the list on the right.. iv i. domin: 0; rnge: b. viii ii. domin: 0; rnge: 0 c. i iii. domin: 0; rnge: d. v iv. domin: 0; rnge: 0 e. ii v. domin: ; rnge: 0 f. vii vi. domin: ; rnge: g. vi vii. domin: ; rnge: 0 h. iii viii. domin: ; rnge: 0. The grph of the inequlit is shded region. Which of the following points lie inside this region? (, 0) (, ) (, ) (, ) (, ) (, 0), (, ), (, ) Remember Wht You Lerned. A good w to remember something is to eplin it to someone else. Suppose ou re studing this lesson with clssmte who thinks tht ou cnnot hve squre root functions becuse ever positive rel number hs two squre roots. How would ou eplin the ide of squre root functions to our clssmte? Smple nswer: To form squre root function, choose either the positive or negtive squre root. For emple, nd re two seprte functions. Chpter 7 0 Glencoe Algebr Answers 7 NAME DATE PERID Stud Guide nd Intervention Squre Root Functions nd Inequlities Squre Root Functions A function tht contins the squre root of vrible epression is squre root function. Emple Grph. Stte its domin nd rnge. Since the rdicnd cnnot be negtive, 0 or. The intercept is. The rnge is 0. Mke tble of vlues nd grph the function. 0 7 Eercises Grph ech function. Stte the domin nd rnge of the function.... D: 0; R: 0 D: 0; R: 0 D: 0; R: 0... D: ; R: 0 D: ;R: 0 D: ;R: 0 Chpter 7 Glencoe Algebr Chpter 7 A9 Glencoe Algebr
10 Answers (Lesson 7) Lesson 77 NAME DATE PERID Stud Guide nd Intervention (continued) Squre Root Functions nd Inequlities Squre Root Inequlities A squre root inequlit is n inequlit tht contins the squre root of vrible epression. Use wht ou know bout grphing squre root functions nd qudrtic inequlities to grph squre root inequlities. Emple Grph. Grph the relted eqution. Since the boundr should be included, the grph should be solid. The domin includes vlues for, so the grph is to the right of. Eercises Grph ech inequlit Chpter 7 Glencoe Algebr NAME DATE PERID 7 Skills Prctice Squre Root Functions nd Inequlities Grph ech function. Stte the domin nd rnge of ech function.... D: 0, R: 0 D: 0, R: 0 D: 0, R: 0... D:, R: 0 D:., R: 0 D:, R: Grph ech inequlit Chpter 7 Glencoe Algebr Chpter 7 A0 Glencoe Algebr
11 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Prctice Squre Root Functions nd Inequlities Grph ech function. Stte the domin nd rnge of ech function.... D: 0, R: 0 D:, R: 0 D:, R: D:,R: 0 D: 7, R: D:,R: Grph ech inequlit RLLER CASTERS The velocit of roller coster s it moves down hill is v v h 0, where v0 is the initil velocit nd h is the verticl drop in feet. If v 70 feet per second nd v 0 8 feet per second, find h. bout 7. ft 90. WEIGHT Use the formul d W E 90, which reltes distnce from Erth d Ws in miles to weight. If n stronut s weight on Erth W E is 8 pounds nd in spce W s is pounds, how fr from Erth is the stronut? bout mi Chpter 7 Glencoe Algebr Answers 7 NAME DATE PERID Word Problem Prctice Squre Root Functions nd Inequlities. SQUARES Cth is building squre roof for her grge. The roof will occup squre feet. Wht re the dimensions of the roof? ft b ft. DISTANCE Lnce is stnding t the side of rod wtching cclist go b. The distnce between Lnce nd the cclist s function of time is given b d 9 t. Grph this function. Find the distnce between Lnce nd the cclist fter seconds.. PENDULUMS The period of pendulum, the time it tkes to complete one swing, is given b the formul p L g where L is the length of the pendulum nd g is ccelertion due to grvit, 9.8 m/s. Find the period of pendulum tht is 0. meters long. Round to the nerest tenth.. seconds 8. m STARS For Eercises 7, use the following informtion. The intensit of the light from n object vries inversel with the squre of the k distnce. In other words, I d.. REFLEXES Rchel nd Ashle re testing one nother s reflees. Rchel drops ruler from given height so tht it flls between Ashle s thumb nd inde finger. Ashle tries to ctch the ruler before it flls through her hnd. The time required to ctch the ruler is d given b t where d is mesured in feet. Complete the tble. Round our nswers to the nerest hundredth. Distnce (in.) Refle Time (seconds). Solve the eqution to find d in terms of I. ki d I. Two strs give off the sme mount of light. However, from Erth their intensities differ. Let I nd I be their intensities nd let d nd d be their respective distnces from Erth. Wht is the rtio of d to d? d I I d I in. 0. in in. 0. in If one str ppers 9 times s intense s the other, how much closer is it to Erth? times s close Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
12 Answers (Lessons 7 nd 7) 7 NAME DATE PERID Enrichment Reding Algebr If two mthemticl problems hve bsic structurl similrities, the re sid to be nlogous. Using nlogies is one w of discovering nd proving new theorems. The following numbered sentences discuss threedimensionl nlog to the Pthgoren theorem. 0 Consider tetrhedron with three perpendiculr fces tht meet t verte. 0 Suppose ou wnt to know how the res A, B, nd C of the three fces tht meet t verte re relted to the re D of the fce opposite verte. 0 It is nturl to epect formul nlogous to the Pthgoren theorem z, which is true for similr sitution in two dimensions. 0 To eplore the threedimensionl cse, ou might guess formul nd then tr to prove it. 0 Two resonble guesses re D A B C nd D A B C. b p z q c Refer to the numbered sentences to nswer the questions.. Use sentence 0 nd the top digrm. The prefi tetr mens four. Write n informl definition of tetrhedron. threedimensionl figure with four fces. Use sentence 0 nd the top digrm. Wht re the lengths of the sides of ech fce of the tetrhedron?, b, nd c;, q, nd r; b, p, nd r; c, p, nd q. Rewrite sentence 0 to stte twodimensionl nlogue. Consider tringle with two perpendiculr sides tht meet t verte C.. Refer to the top digrm nd write epressions for the res A, B, nd C mentioned in sentence 0. Possible nswer: A pr, B pq, C rq. To eplore the threedimensionl cse, ou might begin b epressing, b, nd c in terms of p, q, nd r. Use the Pthgoren theorem to do this. q r, b r p, c p q. Which guess in sentence 0 seems more likel? Justif our nswer. See students eplntions. Chpter 7 Glencoe Algebr r Lesson 77 NAME DATE PERID Lesson Reding Guide nth Roots Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. A bsketbll hs volume of bout 8 cubic inches. Eplin how ou would find the rdius of the bsketbll using clcultor. (Do not ctull clculte the rdius.) Smple nswer: Using clcultor, find the product of times the volume. Divide this number b. Then find the positive cube root result. Round the nswer to the nerest tenth. Red the Lesson. For ech rdicl below, identif the rdicnd nd the inde.. rdicnd: inde: b. rdicnd: inde: c. rdicnd: inde:. Complete the following tble. (Do not ctull find n of the indicted roots.) Number Number of Positive Number of Negtive Number of Positive Number of Negtive Squre Roots Squre Roots Cube Roots Cube Roots Stte whether ech of the following is true or flse.. A negtive number hs no rel fourth roots. true b. represents both squre roots of. true c. When ou tke the fifth root of, ou must tke the bsolute vlue of to identif the principl fifth root. flse Remember Wht You Lerned. Wht is n es w to remember tht negtive number hs no rel squre roots but hs one rel cube root? Smple nswer: The squre of positive or negtive number is positive, so there is no rel number whose squre is negtive. However, the cube of negtive number is negtive, so negtive number hs one rel cube root, which is negtive number. Chpter 7 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
13 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Stud Guide nd Intervention nth Roots Simplif Rdicls Squre Root For n rel numbers nd b, if b, then is squre root of b. nth Root For n rel numbers nd b, nd n positive integer n, if n b, then is n nth root of b.. If n is even nd b 0, then b hs one positive root nd one negtive root. Rel nth Roots of b,. If n is odd nd b 0, then b hs one positive root. n b, b n. If n is even nd b 0, then b hs no rel roots.. If n is odd nd b 0, then b hs one negtive root. Emple Simplif 9z. 8 Emple 9z 8 (7z ) 7z z must be positive, so there is no need to tke the bsolute vlue. Simplif ( ) ( ) [( ) ] ( ) Eercises Simplif p 9 7 p. 0. p 0. m n 9 p m n 7. b 8. b b b 0. (k). 9r. 7p k r p. z. q. 00 z z q 7 0 z p 0 0. not rel number 0.8 p 9. () 8 0. ( ). ( b) b. ( ). (m). (m ) Chpter 7 8 Glencoe Algebr Answers NAME DATE PERID 7 Stud Guide nd Intervention (continued) nth Roots Approimte Rdicls with Clcultor Irrtionl Number number tht cnnot be epressed s terminting or repeting deciml Rdicls such s nd re emples of irrtionl numbers. Deciml pproimtions for irrtionl numbers re often used in pplictions. These pproimtions cn be esil found with clcultor. Emple Approimte 8. with clcultor Eercises Use clcultor to pproimte ech vlue to three deciml plces , , LAW ENFRCEMENT The formul r L is used b police to estimte the speed r in miles per hour of cr if the length L of the cr s skid mrk is mesures in feet. Estimte to the nerest tenth of mile per hour the speed of cr tht leves skid mrk 00 feet long. 77. mi/h 0. SPACE TRAVEL The distnce to the horizon d miles from stellite orbiting h miles bove Erth cn be pproimted b d 8000h. h Wht is the distnce to the horizon if stellite is orbiting 0 miles bove Erth? bout 00 mi Chpter 7 9 Glencoe Algebr Chpter 7 A Glencoe Algebr
14 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Skills Prctice nth Roots Use clcultor to pproimte ech vlue to three deciml plces Simplif (). not rel number s s m 8 n m n. 00p q 0p q. w v 8 w v 7. (c) 9c 8. ( ) b b Chpter 7 0 Glencoe Algebr NAME DATE PERID 7 Prctice nth Roots Use clcultor to pproimte ech vlue to three deciml plces (0.9) Simplif () 0. ( ) not rel number. 9m t 8 m.. r w. () 8 7 m t r w. s 8. m p q s pq 9. m 8 n (m ). ( ) m n m. 9 b 0. ( ) 8. d. 0 7 b 8 ( ) 7d 7. RADIANT TEMPERATURE Therml sensors mesure n object s rdint temperture, which is the mount of energ rdited b the object. The internl temperture of n object is clled its kinetic temperture. The formul T r T k e reltes n object s rdint temperture T r to its kinetic temperture T k. The vrible e in the formul is mesure of how well the object rdites energ. If n object s kinetic temperture is 0 C nd e 0.9, wht is the object s rdint temperture to the nerest tenth of degree? 9.C 8. HER S FRMULA Slvtore is buing fertilizer for his tringulr grden. He knows the lengths of ll three sides, so he is using Hero s formul to find the re. Hero s formul sttes tht the re of tringle is s(s)(s b)(s, c) where, b, nd c re the lengths of the sides of the tringle nd s is hlf the perimeter of the tringle. If the lengths of the sides of Slvtore s grden re feet, 7 feet, nd 0 feet, wht is the re of the grden? Round our nswer to the nerest whole number. ft Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
15 Answers (Lesson 7) Lesson 77 NAME DATE PERID. CUBES Cth is building cubic storge room. She wnts the volume of the spce to be 78 cubic feet. Wht should the dimensions of the cube be? Word Problem Prctice nth Roots ft b ft b ft. MARKUPS A wholesler mnufctures prt for D dollrs. The wholesler sells the prt to deler for P percent mrkup. The deler sells the prt to retiler t n dditionl P percent mrkup. The retiler in turn sells the prt to its customers mrking up the price et nother P percent. Wht is the price tht customers see? If the customer bus the prt for $80 nd the mrkup is 0%, wht pproimtel ws the originl cost to mke the prt? D( P) ; $9.. ASTRNMY A specil form of Kepler s Third Lw of Plnetr Motion is given b P where is the verge distnce of n object from the Sun in AU (stronomicl units) nd P is the period of the orbit in ers. If n object is orbiting the Sun with period of ers, wht is its distnce from the Sun?. AU PENDULUMS For Eercises nd, use the following informtion. Mr. Toplin s phsics clss is eperimenting with pendulums. The clss lerned the L g formul T which reltes the time T tht it tkes for pendulum to swing bck nd forth bsed on grvit g equl to feet per second squred, nd the length of the pendulum L in feet.. TUNING Two notes re n octve prt if the frequenc of the higher note is twice the frequenc of the lower note. Cse is eperimenting with n instrument tht hs notes tuned so tht the frequenc of ech successive note increses b the sme fctor nd the first nd lst note re n octve prt. B wht fctor does the frequenc increse from note to note? or pproimtel.. ne group in the clss mde foot long pendulum. Use the formul to determine how long it will tke for their pendulum to swing bck nd forth..7 seconds. Another group decided the wnted to mke pendulum tht took bout.7 seconds to go bck nd forth. Approimtel how long should their pendulum be?. feet Chpter 7 Glencoe Algebr Answers NAME DATE PERID 7 Enrichment Approimting Squre Roots Consider the following epnsion. ( b ) b b b b b Think wht hppens if is ver gret in comprison to b. The term smll nd cn be disregrded in n pproimtion. ( b ) b is ver b b Suppose number cn be epressed s b, b. Then n pproimte vlue b of the squre root is. You should lso see tht b. b Emple Use the formul b b to pproimte 0 nd b. Let 0 nd b. Let nd b. 0 0 ( 0) ( ) Eercises Use the formul to find n pproimtion for ech squre root to the nerest hundredth. Check our work with clcultor ,, b. Show tht b b for b. b disregrd b ; b b; b b b ; Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
16 Answers (Lesson 7) Lesson 77 NAME DATE PERID Lesson Reding Guide pertions with Rdicl Epressions Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. Describe how ou could use the golden rtio to find the height of golden tringle if ou knew its width. Smple nswer: Use clcultor to multipl the width b nd divide the result b the quntit of. Round this nswer to the nerest tenth. Red the Lesson. Complete the conditions tht must be met for rdicl epression to be in simplified form. The inde n is s smll s possible. The rdicnd contins no fctors (other thn ) tht re nth powers of (n) integer or polnomil. The rdicnd contins no frctions. No rdicls pper in the denomintor... Wht re conjugtes of rdicl epressions used for? to rtionlize binomil denomintors b. How would ou use conjugte to simplif the rdicl epression? Multipl numertor nd denomintor b. c. In order to simplif the rdicl epression in prt b, two multiplictions re necessr. The multipliction in the numertor would be done b the FIL method, nd the multipliction in the denomintor would be done b finding the difference of two squres. Remember Wht You Lerned. ne w to remember something is to eplin it to nother person. When rtionlizing the denomintor in the epression, mn students think the should multipl numertor nd denomintor b. How would ou eplin to clssmte wh this is incorrect nd wht he should do insted. Smple nswer: Becuse ou re working with cube roots, not squre roots, ou need to mke the rdicnd in the denomintor perfect cube, not perfect squre. Multipl numertor nd denomintor b to mke the denomintor 8, which equls. Chpter 7 Glencoe Algebr NAME DATE PERID 7 Stud Guide nd Intervention pertions with Rdicl Epressions Simplif Rdicl Epressions For n rel numbers nd b, nd n integer n : Product Propert of Rdicls. if n is even nd nd b re both nonnegtive, then b n n b. n. if n is odd, then b n n b. n To simplif squre root, follow these steps:. Fctor the rdicnd into s mn squres s possible.. Use the Product Propert to isolte the perfect squres.. Simplif ech rdicl. Quotient Propert of Rdicls For n rel numbers nd b 0, nd n integer n, n n b nb, if ll roots re defined. To eliminte rdicls from denomintor or frctions from rdicnd, multipl the numertor nd denomintor b quntit so tht the rdicnd hs n ect root. Emple Simplif. b 7 Emple b 7 () (b ) b b b 8 Simplif. 8 8 Quotient Propert () Fctor into squres. ( ) () Product Propert ( ) Simplif. Rtionlize the denomintor. 0 Simplif. Eercises Simplif... 9 b 0 b b bb p q pq p Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
17 Answers (Lesson 7) Lesson 77 NAME DATE PERID Stud Guide nd Intervention (continued) pertions with Rdicl Epressions pertions with Rdicls When ou dd epressions contining rdicls, ou cn dd onl like terms or like rdicl epressions. Two rdicl epressions re clled like rdicl epressions if both the indices nd the rdicnds re like. To multipl rdicls, use the Product nd Quotient Properties. For products of the form (b cd) (ef gh), use the FIL method. To rtionlize denomintors, use conjugtes. Numbers of the form b cd nd b cd, where, b, c, nd d re rtionl numbers, re clled conjugtes. The product of conjugtes is lws rtionl number. Emple Simplif Fctor using squres. 0 Simplif squre roots Multipl. 0 0 Combine like rdicls. Emple Simplif ( )( ). ( )( ) 0 Emple Simplif. () () 9 Eercises Simplif ( ). ( 0 ) ( 7)( 7) 8. ( )( ) 9. ( )(0 ) Chpter 7 Glencoe Algebr Answers NAME DATE PERID 7 Skills Prctice Simplif. pertions with Rdicl Epressions b b 7. d f f d 8 f 8.s t s t g z g0gz z. ()(). ()(0) ( )( ) 9. ( )( ) 0. ( 7)( ) 7. ( ) Chpter 7 7 Glencoe Algebr Chpter 7 A7 Glencoe Algebr
18 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Prctice Simplif. pertions with Rdicl Epressions t w t w 8. 8v 8 z v z z 9. 8g k 8 gk k c d 7 c d 9 8 d.. 8 b 9 8b 7. ()() 7. ()(78) ( ) 8 0. ( 7). ( )( ). ( 0)( 0) ( )( 7). ( 7) 7. (08 ) BRAKING The formul s estimtes the speed s in miles per hour of cr when it leves skid mrks feet long. Use the formul to write simplified epression for s if 8. Then evlute s to the nerest mile per hour. 07 ; mi/h. PYTHAGREAN THEREM The mesures of the legs of right tringle cn be represented b the epressions nd 9. Use the Pthgoren Theorem to find simplified epression for the mesure of the hpotenuse. Chpter 7 8 Glencoe Algebr 7 NAME DATE PERID Word Problem Prctice pertions with Rdicl Epressions. CUBES Cth hs rectngulr bo with dimensions 0 inches b inches b 0 inches. She would like to replce it with bo in the shpe of cube but with the sme volume. Wht should the length of side of the cube be? Epress our nswer s rdicl epression in simplest form. 0 8 in. RACING For Eercises nd, use the following informtion nd epress our nswers in simplest form. John is J s ounger brother. The like to rce nd, fter mn rces, the found tht the firest rce ws to run slightl different distnces. The both strt t the sme plce nd run stright for 0. miles. Then the hed for different finishes. In the figure, John nd J s finishing pths re mrked.. PHYSICS The speed of wve trveling t over string is given b where t is u, the tension of the string nd u is the densit. Rewrite the epression in simplest form b rtionlizing the denomintor. tu u. TUNING With ech note higher on pino, the frequenc of the pitch increses b fctor of. Wht is the rtio of the frequencies of two notes tht re steps prt on the pino? Wht is the rtio of the frequencies of two notes tht re 9 steps prt on the pino? Epress our nswers in simplest form. nd 8 Strt 0. mile This time, the tied. Both of them finished the rce in ectl minutes.. If John nd J continued t their verge pces during the rce, ectl how mn minutes would it tke them ech to run mile? Epress our nswer s rdicl epression in simplest form. John: or 0 0 min; J: 0. mile mile John s Finish 0. mile J s Finish. LIGHTS Suppose light hs brightness intensit of I when it is t distnce of d nd brightness intensit of I when it is t distnce of d. These quntities re relted b the eqution d d I I. Suppose I 0 units nd d I units. Wht would d be? Epress our nswer in simplest form. or 0 80 min. Ectl how mn times s fst did John run s J? 0 Chpter 7 9 Glencoe Algebr Chpter 7 A8 Glencoe Algebr
19 Answers (Lessons 7 nd 7) Lesson 77 NAME DATE PERID Enrichment Specil Products with Rdicls Notice tht ()(), or (). In generl, () when 0. Also, notice tht (9)(). In generl, ()() when nd re not negtive. You cn use these ides to find the specil products below. ( b)( b) () (b) b ( b) () b (b) b b ( b) () b (b) b b Emple Find the product: ( )( ). ( )( ) () () Emple Evlute ( 8). ( 8) () 8 (8) 8 () Eercises Multipl.. ( 7)( 7). (0 )(0 ) 8. ( )( ). ( (7)). (000 0) 0. ( )( ) 7. (0 ) ( 0) 0 00 You cn etend these ides to ptterns for sums nd differences of cubes. Stud the pttern below. Then complete Eercises 9. ( 8 )( 8 8 ) ( )( 0 ) 0. ( w)( w w ) w. ( 7 0)( ) 7. ( 8)( 88 8 ) Chpter 7 0 Glencoe Algebr Answers NAME DATE PERID 7 Lesson Reding Guide Rtionl Eponents Get Red for the Lesson Red the introduction to Lesson 7 in our tetbook. The formul in the introduction contins the eponent. Wht do ou think it might men to rise number to the power? Smple nswer: Tke the fifth root of the number nd squre it. Red the Lesson. Complete the following definitions of rtionl eponents. ecept n b For n rel number b nd for n positive integer n, b n when b 0 nd n is even. For n nonzero rel number b, nd n integers m nd n, with n, n n m b m n b n m ( b) n m, ecept when b 0 nd n is even.. Complete the conditions tht must be met in order for n epression with rtionl eponents to be simplified. It hs no negtive eponents. It hs no frctionl eponents in the denomintor. It is not comple frction. The inde of n remining rdicl is the number possible. lest. Mrgrit nd Pierre were working together on their lgebr homework. ne eercise sked them to evlute the epression 7. Mrgrit thought tht the should rise 7 to the fourth power first nd then tke the cube root of the result. Pierre thought tht the should tke the cube root of 7 first nd then rise the result to the fourth power. Whose method is correct? Both methods re correct. Remember Wht You Lerned. Some students hve trouble remembering which prt of the frction in rtionl eponent gives the power nd which prt gives the root. How cn our knowledge of integer eponents help ou to keep this stright? Smple nswer: An integer eponent cn be written s rtionl eponent. For emple,. You know tht this mens tht is rised to the third power, so the numertor must give the power, nd, therefore, the denomintor must give the root. Chpter 7 Glencoe Algebr Chpter 7 A9 Glencoe Algebr
20 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Stud Guide nd Intervention Rtionl Eponents Rtionl Eponents nd Rdicls Definition of b n For n rel number b nd n positive integer n, b n n b, ecept when b 0 nd n is even. Definition of b m n For n nonzero rel number b, nd n integers m nd n, with n, b m n b n m ( b) n m, ecept when b 0 nd n is even. Emple Notice tht Write 8 in rdicl form. Emple Evlute. Notice tht 8 0, 0, nd is odd Eercises Write ech epression in rdicl form Write ech rdicl using rtionl eponents.. 7. b. p 7 b p Evlute ech epression (0.000) Chpter 7 Glencoe Algebr 7 NAME DATE PERID Stud Guide nd Intervention (continued) Rtionl Eponents Simplif Epressions All the properties of powers from Lesson  ppl to rtionl eponents. When ou simplif epressions with rtionl eponents, leve the eponent in rtionl form, nd write the epression with ll positive eponents. An eponents in the denomintor must be positive integers. When ou simplif rdicl epressions, ou m use rtionl eponents to simplif, but our nswer should be in rdicl form. Use the smllest inde possible. Emple Simplif 8. Emple 8 8 ( ) Simplif. ( ) ( ) ( ) ( ) () Eercises Simplif ech epression.... p p 7 0 p. m. 8. s or m s 9 m m p p p or b b 8 b b Chpter 7 Glencoe Algebr Chpter 7 A0 Glencoe Algebr
21 Answers (Lesson 7) Lesson 7 NAME DATE PERID 7 Skills Prctice Rtionl Eponents Write ech epression in rdicl form or ( ) or 8. (s ) s s Write ech rdicl using rtionl eponents Evlute ech epression () Simplif ech epression. 7. c c c 8. m 9 m 9 m 9. q q 0. p p or p p. or. n n n. or. or n n n b 7b Chpter 7 Glencoe Algebr Answers NAME DATE PERID 7 Prctice Rtionl Eponents Write ech epression in rdicl form.... m 7. (n ) or ( ) m 7 or ( m) 7 n n Write ech rdicl using rtionl eponents m n 8. 0 b 79 m n b Evlute ech epression () Simplif ech epression. 8. g 7 g 7 g 9. s s s 0. u u. or. b q or. q t. t b b. b q t t z z z z z b b b 0. ELECTRICITY The mount of current in mperes I tht n pplince uses cn be P R clculted using the formul I, where P is the power in wtts nd R is the resistnce in ohms. How much current does n pplince use if P 00 wtts nd R 0 ohms? Round our nswer to the nerest tenth. 7. mps. BUSINESS A compn tht produces DVDs uses the formul C 88n 0 to clculte the cost C in dollrs of producing n DVDs per d. Wht is the compn s cost to produce 0 DVDs per d? Round our nswer to the nerest dollr. $798 Chpter 7 Glencoe Algebr Chpter 7 A Glencoe Algebr
22 Answers (Lesson 7) Lesson 77 NAME DATE PERID. SQUARING THE CUBE A cube hs side length s. Wht is the side length of the squre tht hs n re equl to the volume of this cube? Write our nswer using rtionl eponents. s Word Problem Prctice Rtionl Eponents. WATER TWER A lrge wter tower stores drinking wter in big sphericl tnk. The mor of the town decides tht the wter tower must be replced with lrger tnk. Town residents insist tht the new tower be sphere. If the new tnk will hold 0 times s much wter s the old tnk, how mn times long should the rdius of the new tnk be compred to the old tnk? Write our nswer using rtionl eponents.. INTEREST Rit opened bnk ccount tht ccumulted interest t the rte of % compounded nnull. Her mone ccumulted interest in tht ccount for 8 ers. She then took ll of her mone out of tht ccount nd plced it into nother ccount tht pid % interest compounded nnull. After ers, she took ll of her mone out of tht ccount. Wht single interest rte when compounded nnull would give her the sme outcome for those ers? Round our nswer to the nerest hundredth of percent..% CELLS For Eercises 7, use the following informtion. The number of cells in cell culture grows eponentill. The number of cells in the culture s function of time is given b the epression N( ) t where t is mesured in hours nd N is the initil size of the culture. 0. After hours, there were 78 cells in the culture. Wht is N? 000. BALLNS A sphericl blloon is being inflted fster nd fster. The volume of the blloon s function of time is 9t. Wht is the rdius of the blloon s function of time? Write our nswer using rtionl eponents. t ( ). How mn cells were in the culture fter 0 minutes? Epress our nswer in simplest form How mn cells were in the culture fter. hours? Epress our nswer in simplest form Chpter 7 Glencoe Algebr 7 NAME DATE PERID Enrichment LesserKnown Geometric Formuls Mn geometric formuls involve rdicl epressions. Mke drwing to illustrte ech of the formuls given on this pge. Then evlute the formul for the given vlue of the vrible. Round nswers to the nerest hundredth.. The re of n isosceles tringle. Two sides hve length ; the other side hs length c. Find A when nd c 7. c A c A 7.0 units. The re of n equilterl tringle with side of length. Find A when 8. A A 7.7 units c. The re of regulr pentgon with side of length. Find A when. A 0 A 7. units. The re of regulr hegon with side of length. Find A when 9. A A 0. units. The volume of regulr tetrhedron with n edge of length. Find V when. V V 0.9 units. The re of the curved surfce of right cone with n ltitude of h nd rdius of bse r. Find S when r nd h. S rr h S. units h 7. Heron s Formul for the re of tringle uses the semiperimeter s, where s b c. The sides of the tringle hve lengths, b, nd c. Find A when, b, nd c. A s(s)(s b)(s c) A units b 8. The rdius of circle inscribed in given tringle lso uses the semiperimeter. Find r when, b 7, nd c 9. r s(s)(s b)(s c) r.9 units s b c c Chpter 7 7 Glencoe Algebr r r Chpter 7 A Glencoe Algebr
23 Answers (Lessons 7 nd 77) Lesson 77 NAME DATE PERID 7 Spredsheet Activit Apprecition nd Deprecition When n sset such s house increses in vlue over time, it is sid to pprecite. If the vlue increses b fied percent ech er, or other period of time, the mount of tht quntit fter t ers is given b ( r) t, where is the initil mount nd r is the percent of increse epressed s deciml. You cn use spredsheet to investigte future vlues of n sset. Emple Michel Blckstock is considering buing piece of investment propert for $9,000. The homes in the re re ppreciting t n verge rte of % per er. Find the epected vlue of the home in er, er nd months, ers, nd ers nd 9 months. Use rows nd to enter the initil mount nd the rte Apprecition.ls of increse. Then use Column A to enter the mounts of time. Enter the numbers of months s frction of er A B Initil vlue = $9, since t is mesured in ers. Column B contins the Rte = 0.0 formuls for the vlue of the home. Yers Vlue $98, Formt the cells contining the vlues s currenc so. $00,7. tht the re displed s dollrs nd cents. The epected 7 $,. vlue of the home fter ech mount of time is shown $,79.7 in the spredsheet. Sheet Sheet Eercises. If Mr. Blckstock chooses nother propert in the neighborhood tht costs $99,900, wht re the epected vlues of tht home in the sme periods of time? $0,89.00, $0,9., $,88.87, $0, Wht would Mr. Blckstock s profit be on the $99,900 home if he sold it fter 9 ers nd months? $, If n ntique chir worth $.00 increses in vlue n verge of % ever er, how much will it be worth net er? $ ften ssets like crs decrese in vlue over time. This sset is sid to deprecite. If the vlue decreses b fied percent ech er, or other period of time, the mount of tht quntit fter t ers is given b ( r) t, where is the initil mount nd r is the percent of decrese epressed s deciml. Use spredsheet to find the vlue of cr purchsed for $8,00 fter ers, ers nd months, nd ers nd months if the cr deprecites t rte of % per er. Chpter 7 8 Glencoe Algebr Answers 77 NAME DATE PERID Lesson Reding Guide Solving Rdicl Equtions nd Inequlities Get Red for the lesson Red the introduction to Lesson 77 in our tetbook. Eplin how ou would use the formul in our tetbook to find the cost of producing,000 computer chips. (Describe the steps of the clcultion in the order in which ou would perform them, but do not ctull do the clcultion.) Smple nswer: Rise,000 to the power b tking the cube root of,000 nd squring the result (or rise,000 to the power b entering,000 ^ (/) on clcultor). Multipl the number ou get b 0 nd then dd 00. Red the Lesson.. Wht is n etrneous solution of rdicl eqution? Smple nswer: number tht stisfies n eqution obtined b rising both sides of the originl eqution to higher power but does not stisf the originl eqution b. Describe two ws ou cn check the proposed solutions of rdicl eqution in order to determine whether n of them re etrneous solutions. Smple nswer: ne w is to check ech proposed solution b substituting it into the originl eqution. Another w is to use grphing clcultor to grph both sides of the originl eqution. See where the grphs intersect. This cn help ou identif solutions tht m be etrneous.. Complete the steps tht should be followed in order to solve rdicl inequlit. Step If the inde of the root is even, identif the vlues of the vrible for which the rdicnd is nonnegtive. Step Solve the inequlit lgebricll. Step Test vlues to check our solution. Remember Wht You Lerned. ne w to remember something is to eplin it to nother person. Suppose tht our friend Leor thinks tht she does not need to check her solutions to rdicl equtions b substitution becuse she knows she is ver creful nd seldom mkes mistkes in her work. How cn ou eplin to her tht she should nevertheless check ever proposed solution in the originl eqution? Smple nswer: Squring both sides of n eqution cn produce n eqution tht is not equivlent to the originl one. For emple, the onl solution of is, but the squred eqution hs two solutions, nd. Chpter 7 9 Glencoe Algebr Chpter 7 A Glencoe Algebr
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