Ratio, Proportion, and Variation

Transcription

1 7.3 Rtio, Proportion, nd Vrition Rtio One of the most frequently used mthemticl concepts in everydy life is rtio. A bsebll plyer s btting verge is ctully rtio. The slope, or pitch, of roof on building my be expressed s rtio. Rtios provide wy of compring two numbers or quntities. Rtio A rtio is quotient of two quntities. The rtio of the number to the number b is written to b, or b. b, When rtios re used in compring units of mesure, the units should be the sme. This is shown in Exmple 1.

2 340 CHAPTER 7 The Bsic Concepts of Algebr I Imge on film When you look long wy down stright rod or rilrod trck, it seems to nrrow s it vnishes in the distnce. The point where the sides seem to touch is clled the vnishing point. The sme thing occurs in the lens of cmer, s shown in the figure. Suppose I represents the length of the imge, O the length of the object, d the distnce from the lens to the film, nd D the distnce from the lens to the object. Then or Vnishing point d D Lens I O d D. O Object Imge length Imge distnce Object length Object distnce Given the length of the imge on the film nd its distnce from the lens, then the length of the object determines how fr wy the lens must be from the object to fit on the film. EXAMPLE 1 Write rtio for ech word phrse. () the rtio of 5 hours to 3 hours This rtio cn be written s 53. (b) the rtio of 5 hours to 3 dys First convert 3 dys to hours: 3 dys hours. The rtio of 5 hours to 3 dys is thus 572. Proportion Proportion We now define specil type of eqution clled proportion. A proportion is sttement tht sys tht two rtios re equl. For exmple, is proportion tht sys tht the rtios 34 nd 1520 re equl. In the proportion b c d,, b, c, nd d re the terms of the proportion. The nd d terms re clled the extremes, nd the b nd c terms re clled the mens. We cn red the proportion s is to b s c is to d. Beginning with this proportion nd multiplying both b c d sides by the common denomintor, bd, gives bd b bd c d d bc. Tht is, the product of the extremes equls the product of the mens. The products d nd bc cn lso be found by multiplying digonlly. bc b c d d This is clled cross multipliction nd d nd bc re clled cross products. Cross Products If, then the cross products d nd bc re equl. b c d Also, if d bc, then (s long s b 0, d 0). b c d

3 7.3 Rtio, Proportion, nd Vrition 341 From the rule given on pge 340, if then However, if then c b b c d d bc. d, d cb, or d bc. This mens tht the two proportions re equivlent nd the proportion b c d cn lso be written s c b d. Sometimes one form is more convenient to work with thn the other. Four numbers re used in proportion. If ny three of these numbers re known, the fourth cn be found. EXAMPLE 2 Solve ech proportion. () (b) 63 x 9 5 The cross products must be equl. The solution set is r 8r r x 315 9x 35 x Cross products Divide by 9. Set the cross products equl. r Divide by 8; express in lowest terms. The solution set is EXAMPLE 3 Solve the eqution Find the cross products, nd set them equl to ech other. m 11 2 m 2 5 3m 2 5m 1 3m 6 5m 5 3m 5m 11 2m 11 m 1. 3 Be sure to use prentheses. Distributive property Add 6. Subtrct 5m. Divide by 2. The solution set is 11. 2

4 342 CHAPTER 7 The Bsic Concepts of Algebr While the cross product method is useful in solving equtions of the types found in Exmples 2 nd 3, it cnnot be used directly if there is more thn one term on either side. For exmple, you cnnot use the method directly to solve the eqution 4 x 3 1 9, becuse there re two terms on the left side. EXAMPLE 4 Biologists use lgebr to estimte the number of fish in lke. They first ctch smple of fish nd mrk ech specimen with hrmless tg. Some weeks lter, they ctch similr smple of fish from the sme res of the lke nd determine the proportion of previously tgged fish in the new smple. The totl fish popultion is estimted by ssuming tht the proportion of tgged fish in the new smple is the sme s the proportion of tgged fish in the entire lke. Suppose biologists tg 300 fish on My 1. When they return on June 1 nd tke new smple of 400 fish, 5 of the 400 were previously tgged. Estimte the number of fish in the lke. Let x represent the number of fish in the lke. Set up nd solve proportion. Tgged fish on My 1 Totl fish in the lke l 300 l x There re pproximtely 24,000 fish in the lke. Unit pricing deciding which size of n item offered in different sizes produces the best price per unit uses proportions. Suppose you buy 36 ounces of pncke syrup for $3.89. To find the price per unit, set up nd solve proportion. 36 ounces 1 ounce 5x 120,000 x 24, x k Tgged fish in the June 1 smple k Totl number in the June 1 smple 36x 3.89 Cross products x 3.89 Divide by x.108 Use clcultor. Thus, the price for 1 ounce is $.108, or bout 11 cents. Notice tht the unit price is the rtio of the cost for 36 ounces, $3.89, to the number of ounces, 36, which mens tht the unit price for n item is found by dividing the cost by the number of units. Size Price 36-ounce $ ounce $ ounce $1.89 EXAMPLE 5 Besides the 36-ounce size discussed erlier, the locl supermrket crries two other sizes of populr brnd of pncke syrup, priced s shown in the tble. Which size is the best buy? Tht is, which size hs the lowest unit price? To find the best buy, divide the price by the number of units to get the price per ounce. Ech result in the tble on the next pge ws found by using clcultor nd rounding the nswer to three deciml plces.

5 7.3 Rtio, Proportion, nd Vrition 343 Size 36-ounce 24-ounce 12-ounce Unit Cost (dollrs per ounce) $ $ $ $.108 kthe best buy $.116 $.158 Number of Rooms Cost of the Job 1 $ $ $ $ $ Since the 36-ounce size produces the lowest price per unit, it would be the best buy. (Be creful: Sometimes the lrgest continer does not produce the lowest price per unit.) Vrition Suppose tht crpet clening service chrges $22.50 per room to shmpoo crpet. The tble in the mrgin shows the reltionship between the number of rooms clened nd the cost of the totl job for 1 through 5 rooms. If we divide the cost of the job by the number of rooms, in ech cse we obtin the quotient, or rtio, (dollrs per room). Suppose tht we let x represent the number of rooms nd y represent the cost for clening tht number of rooms. Then the reltionship between x nd y is given by the eqution y x or y 22.50x. This reltionship between x nd y is n exmple of direct vrition. Direct Vrition y vries directly s x, oryis directly proportionl to x, if there exists nonzero constnt k such tht y kx y or, equivlently, x k. The constnt k is numericl vlue clled the constnt of vrition. EXAMPLE 6 Suppose y vries directly s x, nd y 50 when x 20. Find y when x 14. Since y vries directly s x, there exists constnt k such tht y kx. Find k by replcing y with 50 nd x with 20. y kx 50 k k

6 344 CHAPTER 7 The Bsic Concepts of Algebr Since y kx nd k 52, y 5 2 x. Now find y when x 14. y The vlue of y is 35 when x d FIGURE 5 EXAMPLE 7 Hooke s lw for n elstic spring sttes tht the distnce spring stretches is directly proportionl to the force pplied. If force of 150 pounds stretches certin spring 8 centimeters, how much will force of 400 pounds stretch the spring? See Figure 5. If d is the distnce the spring stretches nd f is the force pplied, then d kf for some constnt k. Since force of 150 pounds stretches the spring 8 centimeters, d kf 8 k 150 k , Find k. Formul d 8, f 150 nd d 4. For force of 400 pounds, 75 f d Let f 400. The spring will stretch 643 centimeters if force of 400 pounds is pplied. In summry, follow these steps to solve vrition problem. Solving Vrition Problem Step 1: Write the vrition eqution. Step 2: Substitute the initil vlues nd solve for k. Step 3: Rewrite the vrition eqution with the vlue of k from Step 2. Step 4: Substitute the remining vlues, solve for the unknown, nd find the required nswer. In some cses one quntity will vry directly s power of nother. Direct Vrition s Power y vries directly s the nth power of x if there exists rel number k such tht y kx n.

7 7.3 Rtio, Proportion, nd Vrition 345 r A = πr 2 An exmple of direct vrition s power involves the re of circle. The formul for the re of circle is A r 2. Here, is the constnt of vrition, nd the re vries directly s the squre of the rdius. EXAMPLE 8 The distnce body flls from rest vries directly s the squre of the time it flls (here we disregrd ir resistnce). If skydiver flls 64 feet in 2 seconds, how fr will she fll in 8 seconds? Step 1: If d represents the distnce the skydiver flls nd t the time it tkes to fll, then d is function of t, nd for some constnt k. Step 2: To find the vlue of k, use the fct tht the object flls 64 feet in 2 seconds. d kt 2 64 k2 2 k 16 Step 3: With this result, the vrition eqution becomes d 16t 2. d kt 2 Formul Let d 64 nd t 2. Find k. Step 4: Now let t 8 to find the number of feet the skydiver will fll in 8 seconds. d The skydiver will fll 1024 feet in 8 seconds. Let t 8. In direct vrition where k 0, s x increses, y increses, nd similrly s x decreses, y decreses. Another type of vrition is inverse vrition. Inverse Vrition y vries inversely s x if there exists rel number k such tht y k x, or, equivlently, xy k. Also, y vries inversely s the nth power of x if there exists rel number k such tht y k x n.

8 346 CHAPTER 7 The Bsic Concepts of Algebr Johnn Kepler ( ) estblished the importnce of the ellipse in 1609, when he discovered tht the orbits of the plnets round the sun were ellipticl, not circulr. The orbits of the plnets re nerly circulr. Hlley s comet, which hs been studied since 467 B.C., hs n ellipticl orbit which is long nd nrrow, with one xis much longer thn the other. This comet ws nmed for the British stronomer nd mthemticin Edmund Hlley ( ), who predicted its return fter observing it in The comet ppers regulrly every 76 yers. EXAMPLE 9 The weight of n object bove the erth vries inversely s the squre of its distnce from the center of Erth. A spce vehicle in n ellipticl orbit hs mximum distnce from the center of Erth (pogee) of 6700 miles. Its minimum distnce from the center of Erth (perigee) is 4090 miles. See Figure 6 (not to scle). If n stronut in the vehicle weighs 57 pounds t its pogee, wht does the stronut weigh t the perigee? Spce vehicle t perigee d 2 Erth FIGURE 6 d 1 Spce vehicle t pogee If w is the weight nd d is the distnce from the center of Erth, then for some constnt k. At the pogee the stronut weighs 57 pounds nd the distnce from the center of Erth is 6700 miles. Use these vlues to find k. 57 k k w k d 2 Let w 57 nd d Then the weight t the perigee with d 4090 miles is w pounds. Use clcultor It is common for one vrible to depend on severl others. For exmple, if one vrible vries s the product of severl other vribles (perhps rised to powers), the first vrible is sid to vry jointly s the others. EXAMPLE 10 The strength of rectngulr bem vries jointly s its width nd the squre of its depth. If the strength of bem 2 inches wide by 10 inches deep is 1000 pounds per squre inch, wht is the strength of bem 4 inches wide nd 8 inches deep? If S represents the strength, w the width, nd d the depth, then S kwd 2 for some constnt k. Since S 1000 if w 2 nd d 10, 1000 k Let S 1000, w 2, nd d 10.

9 7.3 Rtio, Proportion, nd Vrition 347 Solving this eqution for k gives so 1000 k k k 5, S 5wd 2. Find S when w 4 nd d 8 by substitution in S 5wd 2. S Let w 4 nd d 8. The strength of the bem is 1280 pounds per squre inch. There re mny combintions of direct nd inverse vrition. The finl exmple shows typicl combined vrition problem. 9 m 1 m Lod = 8 metric tons EXAMPLE 11 The mximum lod tht cylindricl column with circulr cross section cn hold vries directly s the fourth power of the dimeter of the cross section nd inversely s the squre of the height. A 9-meter column 1 meter in dimeter will support 8 metric tons. See Figure 7. How mny metric tons cn be supported by column 12 meters high nd 23 meter in dimeter? Let L represent the lod, d the dimeter, nd h the height. Then FIGURE 7 L kd 4 h 2. k Lod vries directly s the 4th power of the dimeter. k Lod vries inversely s the squre of the height. Now find k. Let h 9, d 1, nd L 8. 8 k k 81 k 648 h 9, d 1, L 8 Substitute 648 for k in the first eqution. L 648d4 h 2 Now find L when h 12 nd d L Let h 12, d 2 3. The mximum lod is 89 metric ton.