10.6 Applications of Quadratic Equations


 Allan McDowell
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1 10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given, flling object problems, problems involving geometric shpes. Just to nme few. There re mny other types of ppliction problems tht use qudrtic equtions, however, we will concentrte on these types to simplify the mtter. We must be very creful when solving these problems since sometimes we wnt the mimum or minimum of the qudrtic, nd sometimes we simply wnt to solve or evlute the qudrtic. For the problems where we wnt to find the mimum or minimum vlue, we recll from the lst section tht these vlues lwys occur t the verte. Therefore, we will need to find the verte. When deling with word problems it is generlly esier nd more efficient to use the b method to find the verte. Now lets see some emples. Emple 1: The number of bcteri in refrigerted food is given by N ( T ) 0T 0T + 10, for T 14 nd where T is the temperture of the food in Celsius. At wht temperture will the number of bcteri be miniml? Solution: First we cn see tht we hve qudrtic function given to us. Moreover, the prbol would open up. So tht mens tht the verte of the prbol represents minimum vlue. This mens the first thing we need to do is determine the verte, since we wnt the miniml number of bcteri. As we stted, we will use the formul b to find the verte since it is much more efficient thn the completing the squre method for finding the verte in word problems. Since our function is in terms of T, the formul we relly use is T b. So we get T ( 0) So the T vlue of the verte is ½. Now we must look bck t the question to see wht we relly wnted. Since we wnt the temperture t which the number is minimum nd T is the temperture in Celsius, this is the vlue we wnt. Therefore, the minimum number of bcteri re present when the temperture is ½ 0 Celsius. Notice in the lst emple we only wnted the vlue of the verte. Tht is, the vlue for which the verte occurs. If the question ws sking for wht the mimum or minimum vlue ws, we would hve hd to find the y vlue of the verte. Alwys remember, the vlue refers to the vlue t which the function reches the mimum or minimum, the y vlue refers to wht the vlue of tht mimum or minimum ctully is.
2 Emple : The height, h, in feet of n object bove the ground is given by h + 64t + 190, t 0 where t is the time in seconds. Find the time it tkes the object to strike the ground nd find the mimum height of the object. Solution: Let s first find the time it tkes for the object to hit the ground. Since represents the height bove the ground, we would like to know t wht time h 0. So in the eqution h + 64t we will set h 0 nd solve for the time, t. We hve h + 64t t So we simply wnt to solve this qudrtic eqution. It is esiest to use the qudrtic formul in this sitution. So we get 64 ± t 64 ± ± ± , ( 16)( 190) ( 16) 1656 However, since t represents time, we must throw out Therefore, it tkes 5.98 seconds for the object to strike the ground. The other prt of the question is we wnt to know tht mimum height tht the object reches. Since we cn see tht the function is clerly qudrtic function which opens down, we know tht this mimum must occur t the verte. So let s find the verte. Agin we use the formul t b. So we hve t t 64 ( 16) 64 So t seconds the object reches its mimum height. However, we wnted to know wht tht mimum height is. Therefore, we must find the vlue of the verte, in this cse it will be the vlue of h when t. So we plug this in to get h So the mimum height is 54 feet. ( ) ( ) h
3 Emple : The length of rectngle is three more thn twice the width. Determine the dimensions tht will give totl re of 7 m. Wht is the minimum re tht this rectngle cn hve? Solution: First we need to drw picture to visulize the problem. Since the length is more tht twice the width, we will hve l w +. So we hve the following picture w l w + Now, since both prts of this question del with the re of this rectngle, lets begin by generting function for the re. Since A l w we hve A w + w ( ) w + w For the first prt, we wnt to know wht dimensions mke n re of 7 m. Thus, we cn insert 7 for A into our function nd solve for w. We hve w + w 7 0 ( w )( w + 9) 7 w 0 w, + w So, since w represents the width of rectngle we must omit the negtive vlue. Therefore, we hve w. Plugging tht vlue into l w + we get l + 9. Therefore, the dimensions tht give the rectngle n re of 7 m re 9 m by m. The second prt of the problem sks us for the minimum possible re. So going bck to our re function, A w + w, we see tht the prbol opens up nd therefore the minimum would occur t the verte. So we find the verte using w b like usul. We get w 4 But remember, w represents the width of rectngle. So, since the minimum occurs when the width is negtive, the rectngle must hve no minimum re. So we simply sy no minimum re eists for this given sitution. 9
4 Emple 4: Two rectngulr corrl s re to be mde from 100 yds of fencing s seen below. If the rncher wnts the totl re to be mimum, wht dimensions should be used to mke the corrl s? Solution: Since we re looking to mimize the re, we need to generte function for the re so tht we will only need to find the mimum of this re function. The first thing we should do is decide on some lbels nd vribles. Lets cll the longer side nd the shorter side y. This gives the following picture: y The first thing we must do is epress y in terms of, tht wy we will only hve to del with one vrible. So the totl mount of fencing to be used cn be represented by +++y+y. But we know the totl mount to be used is 100yds. 100 Therefore, 100+y. So y y Thus we hve the picture 100 So, the totl re of the rectngle is A ( ) 100 A l w 100 Now tht we hve function for the re we need to find the mimum, if one eists. Notice tht we cn write our re function s
5 ( ) A 100 ( 50 ) + 50 We cn see tht our function is qudrtic function tht opens down. Thus, the verte will indeed be mimum. So we proceed in finding the verte. Agin we use the b pproch. 50 b 50 ( ) 50 So the vlue of the verte is t 50 or 16. Since represented the longer side, we know tht the mimum re occurs when longer side hs length 16 ft. Now looking bck t the question, we wnt the dimensions tht will produce the mimum re, not the vlue of the mimum re. 50 A we would be finding the vlue of the mimum re, since A is the re If we were to find ( ) function. However, recll from the picture we mde erlier tht 50 ( ) 100 y y. Therefore, So the y vlue t which the mimum re occurs is 5 ft. So our picture is slightly off since y is relly the longer side. However, we only need to picture to generte our lgebric epressions nd equtions. So, the dimensions tht will produce the mimum re re 16 ft by 5 ft. Agin, we need to be very creful to find ectly wht the sitution is sking for. Sometimes the problem is sking for vlue of the verte, sometimes the problem is sking for the solutions to the qudrtic nd sometimes the problem is merely sking to evlute qudrtic function. We must crefully red ech question to determine ectly wht is being sked Eercises 1. The number of bord feet in 16 foot long tree is pproimted by the model F ( d ) 0.77d 1.d 9. 1 where F is the number of feet nd d is the dimeter of the log. How mny bord feet re in log with dimeter 1 inches? Wht is the dimeter tht will produce the minimum number of bord feet?
6 . The number of horsepower needed to overcome wind drg on certin utomobile is given N() s 0.005s s by, where s is the speed of the cr in miles per hour. How much horsepower is needed to overcome the wind drg on this cr if it is trveling 50 miles per hour? At wht speed will the cr need to use 00 horsepower to overcome the wind drg?. The number of bsebll gmes tht must be scheduled in legue with n tems is given by ( ) G n n n where ech tem plys every other tem ectly once. A legue schedules 15 gmes. How mny tems re in the legue? 4. For the yers of 198 to 1990, the number of mountin bike owners m (in millions) in the US m 0.7t.65t +.96, t 10 t cn be pproimted by the model where represents 198. In which yer did.5 million people own mountin bikes? In wht yer ws the number of mountin bike owners t minimum? C where C is the totl cost in dollrs nd is the number of tennis blls produced. Wht number of tennis blls will produce the minimum? 5. A mnufcturer of tennis blls hs dily cost of ( ) t t, where v 6. The vlue of Jennifer s stock portfolio is given by the function ( ) is the vlue of the portfolio in hundreds of dollrs nd t is the time in months. How much money did Jennifer strt with? When will the vlue of Jennifer s portfolio be t mimum? v t t + t where v is the vlue of the portfolio in hundreds of dollrs nd t is the time in months. How much money did Jon strt with? Wht is the minimum vlue of Jon s portfolio? 7. The vlue of Jon s stock portfolio is given by the function ( ) 8. Find the number of units tht produce the mimum revenue R , where R is the totl revenue (in dollrs) nd is the number of units sold. 9. A tetile mnufcturer hs dily production costs of, where C is the totl cost (in dollrs) nd is the number of units produced. How mny units should be produced ech dy to yield minimum cost? v t C 10, C , 10. A mnufcturer of light fitures hs dily production costs of ( ) where C is the totl cost (in dollrs) nd is the number of units produced. How mny units should be produced every dy to yield minimum cost? 11. A compny s weekly revenue in dollrs is given by ( ) R 000, where is the number of items produced during week. Wht mount of items will produce the mimum revenue? 1. A compny erns weekly profit of P dollrs by selling items, ccording to the eqution P ( ) mimize the profit?. How mny items does the compny hve to sell ech week to
7 1. Advertising revenue for newsppers in the United Sttes for the yers 1985 through 1999 is pproimted by the model R t 0.8t where R is revenue in billions of dollrs nd t represents the yer with t 5 corresponding to the yer In wht yer will revenue be mimum? 14. A bll rolls down slope nd trvels distnce the distnce is 17 feet. t d 6t + h t 15. The height in feet of bottle rocket is given by ( ) feet in t seconds. Find when 160t where t is the time in seconds. How long will it tke for the rocket to return to the ground? Wht is the height fter seconds? 16. A foul bll leves the end of bsebll bt nd trvels ccording to the formul h () t 64t where h is the height of the bll in feet nd t is the time in seconds. How long will it tke for the bll to rech height of 64 feet in the ir? 17. A model rocket is projected stright upwrd from the ground level ccording to the height h + 19t, t > 0 eqution, where h is the height in feet nd t is the time in seconds. At wht time is the height of the rocket mimum nd wht is tht height? 18. Emm hits golf bll of the tee. The height of the bll is given by y where y is the height in yrds bove the ground nd is the horizontl distnce from the tee in yrds. How fr does Emm hit the bll? Wht is the mimum height of the bll? 19. Jon is hitting bseblls. When he tosses the bll into the ir, his hnd is 5 feet bove the ground. He hits the bll when it flls bck to height of 4 feet. The height of the bll is given by h 5 + 5t, where t is in seconds. How much time will pss before Jon hits the bll? Wht is the mimum height the bll ttins? 0. The height h in feet of projectile lunched verticlly upwrd from the top of 96foot tll tower when time t 0 is given by h t. How long will it tke the projectile to strike the ground? Wht is the mimum height tht the projectile reches? 1. The formul h + 48t gives the height of n object thrown from building 160 feet high with n initil speed of 48 ft/sec, where t is mesured in seconds. Find the time for the object to hit the ground nd find the mimum height of the object.. While plying bsketbll this weekend Frnk shoots n irbll. The height in feet of the bll is given by h + t + 8. How long will it tke the bll to strike the ground? Wht is the mimum height of the bll?. While on n Audubon field trip Jennifer sees RedTil Hwk drop its prey. The height in feet of the prey is given by h + 48t How long will it tke the prey to strike the ground? Wht is the mimum height of the prey? h h
8 4. While plying ctch with his grndson yesterdy Tim throws bll s hrd s possible into the ir. The height h in feet of the bll is given by h + 64t + 8, where t is in seconds. How long will it tke until the bll reches the grndson s glove if he ctches it t height of feet? Wht is the mimum height of the bll? The pth of high diver is given by y where y is the height in feet 9 9 bove the wter nd is the horizontl distnce from the end of the diving bord in feet. Wht is the mimum height of the diver nd how fr out from the end of the diving bord is the diver when he hits the wter? 6. The height h in feet of projectile lunched verticlly upwrd from the top of 96foot tll bridge is given by h t where t is time in seconds. Wht is the mimum height nd how long will it tke the projectile to strike the ground? 7. The height h in feet of projectile lunched verticlly upwrd from the top of foot tll bridge is given by h t where t is time in seconds. When does the projectile rech mimum height nd how long will it tke the projectile to strike the ground? 8. The height h in feet of projectile lunched verticlly upwrd from the top of 80foot tll bridge is given by h t where t is time in seconds. When does the projectile rech mimum height nd how long will it tke the projectile to strike the ground? 9. The length of rectngulr flower grden is 5 feet more thn its width. If the re of the grden is 104 squre feet, find the dimensions of the flower grden. 0. The height of tringulr flower grden is 6 feet more thn the length of the bse. If the re of the grden is 8 squre feet, find the dimensions of the flower grden. 1. The length of rectngulr plot of lnd is 10 yrds more thn its width. If the re of the lnd is 600 squre yrds, find the dimensions of the plot of lnd.. The length of rectngulr plot of lnd is 10 yrds more thn its width. If the re of the lnd is 600 squre yrds, find the dimensions of the plot of lnd.. The height of tringulr window is feet less thn its bse. If the re of the window is 0 squre feet, find the dimensions of the window. 4. The length of PingPong tble is ft more thn twice the width. The re of PingPong tble is 90 squre feet. Wht re the dimensions of PingPong tble? 5. The width of rectngle is five less thn twice the length. Wht is the minimum re of such rectngle? 6. The length of rectngle is one more thn the width. Wht is the minimum re of such rectngle? 7. The bse of tringle is one more thn four times the height. Determine the dimensions tht will give totl re of 9 cm. Wht is the minimum re of such tringle? 8. The height of tringle is two more thn three times the bse. Determine the dimensions tht will give totl re of 8 yds. Wht is the minimum re of such tringle?
9 9. The perimeter of rectngle is 50 yds. Wht re the dimensions tht will produce the mimum re of such rectngle? 40. The perimeter of rectngle is 70 m. Wht re the dimensions tht will produce the mimum re of such rectngle? 41. The perimeter of rectngle is 100 ft. Wht is the mimum re of such rectngle? 4. The perimeter of rectngle is 10 cm. Wht is the mimum re of such rectngle? 4. Three hundred feet of fencing is vilble to enclose rectngulr yrd long side of the St. John s River, which is one side of the rectngle s seen below. Wht dimensions will produce n re of 10,000 ft? Wht is the mimum re tht cn be enclosed? 44. Five hundred feet of fencing is vilble to enclose rectngulr lot long side of highwy 65. Cl Trns will supply the fencing for the side long the highwy, so only three sides re needed s seen below. Wht dimensions will produce n re of 40,000 ft? Wht is the mimum re tht cn be enclosed? 45. Two rectngulr pens re to be mde from 00 yds of fencing s seen below. Determine the dimensions tht will produce the mimum re. 46. Two rectngulr lots re to be mde from 400 ft of fencing s seen below. Determine the dimensions tht will produce the mimum re.
10 47. Three rectngulr corrls re to be mde from 800 meters of fencing s seen below. Determine the dimensions tht will produce the mimum re. Wht is re of one of the corrls? 48. Three rectngulr corrls re to be mde from 100 yrds of fencing s seen below. Determine the dimensions tht will produce the mimum re. Wht is re of one of the corrls? 49. John V. wnts to fence three sides of rectngulr eercise yrd for his dog. The fourth side of the eercise yrd will be side of the house. He hs 80 feet of fencing vilble. Find the dimensions of the eercise yrd tht will produce the mimum enclosed re. 50. Chrles wnts to build vegetble grden such tht three sides of the grden re fenced nd the fourth side of the grden will be the eisting bck fence. He hs 0 feet of fencing vilble. Find the dimensions of the grden tht will produce the mimum enclosed re. 51. Chris wnts to mke n enclosed rectngulr re for mulch pile. She wnts to mke the enclosure in such wy s to use corner of her bck yrd. She lso wnts it to be twice s long s it is wide. Since the yrd is lredy fenced, she simply needs to construct two sides of the mulch pile enclosure. She hs only 15 feet of mteril vilble. Find the dimensions of the enclosure tht will produce the mimum re. 5. Pul, rncher, hs 00 ft. of fencing to enclose two djcent rectngulr corrls. Wht re the dimensions tht will produce the mimum enclosed re? 5. Ptrick hs 1400 ft. of irrigtion piping he wnts to use to irrigte his bck lwn. He wnts to ly the piping in such mnor s to cut off three equl sized rectngulr regions in the yrd. Wht re the dimensions tht will produce the mimum enclosed re? 54. Show tht mong ll rectngles of fied perimeter p the one with the lrgest re is squre. 55. An Athletic field with perimeter of 4 1 mile consists of rectngle with semicircle t ech end, s shown below. Find the dimensions tht yield the gretest possible re for the rectngulr region. r
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