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# Note for MATH 1310 at UH 2

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Secl39nn m Direcl Variarnn Inverse Variarnmna Juianzr39n nn Page I or 9 DirectVariation Inverse Variation and Joint Variation Direct Variation P is dlreetly proportlonal to Q is expressed by the equatlon P kg where k is a nonzero eonstant We also say thatP l5 proportlonal to Q orP vanes dlrectly as Q Inverse Variatlnn P ls mversely propomonal to Q ls expressed bythe equatron P e where k ls a nonzero eonstant We also say thatP vanes mversely as Q JnlntVarlatlnn P ls Jomtly propomonal to Q and R ls expressed bythe equatron PkQR where k ls a nonzero eonstant We also say thatP vanes Jomtly as Q and R The eonstantk ls ealled the ennstant or prnpnninnality Example 1 M ls mversely propomonal to N a Wnte aformulato express the glven staternent b EN 6 then M 2 Use the formulaln part a andthe glven lnforrnatlon to nd the eonstant of proporaonahty Snlntlnn to Example 1 Part a Letk be the eonstant ofpropomonallty M ls lnversely propomonal to le expressed by the equatron below M N Secl39nn m Direcl Vnrizrnn Inverse vmmhmm JuinIVzr39n nn Pig 2 hr 9 12m h To nd It subshtute N s andM 2 rnto the formulaM 26 wm woww 12 The eonstant ofproporuonahty rs 1t 12 Example 2 S rs drreet1y propomonal to the square ofT a Wnte aformulato express the grven staternent b If T 8 then S 14 Use the formulam pan a andthe grven mfor mauon to nd the eonstant of propomonahty Snllltinn tn Example 2 Part a Letk be the eonstant ofpropomonahty S rs drreeuy proportional to the square of T rs expressed by the equahon below S kTZ 12m h To nd It subshtute T 8 and S 14 rnto the formulaS kT2 14 Mg 14 64k 14 Mk 5 V 7 E The eonstant of propomonahty rs k Secl39nn m Dilecl Variathn lttvetse Variathnatttl Juianzr39n nn Page 3 hr 9 Example 3 A lsjolndy ptopomonal to 3 and c a whte afotmttlato exptess the gtveh statement b KB 0 andC9 thehA 34 Use the fomulaln pan a and the gtveh mfol rnatlon to that the constant of ptopotttonaltty Snllltinn tn Example Part a Letk be the constant ofpropomonallty A ts totmly propomonal to 3 and 5 ts expressed by the equahon below kBC Part h To that k subshtttte 3 10 c e 9 andA 34 thto the formulaA 184 34 k109 34 90k a 96k 90 E 45 The constant of ptopotttonaltty ts k e Example 4 A ts atteetly proponlonal to 3 mdlnversely proponlonal to c whte aformula to exptess the gtveh statement Snllltinn tn Example 4 Secl39nn 3 Direct Variarhm lhverse Varlalhmml Juianzr39n nn Page t at 9 Letk be the constant ofpropomonallty A ls duecdy proporhorral to 3 and lnversely propomonal to c ls expressed by the equatror below Ae 5 Example 5 S ls mversely propomonal to x a whte aformulato express the glven statemeht b Ifx 9 then s 2 Use the formulam part a andthe glven lnformatlon to nd the constant of propornonallty Snllltinn tn Example 5 Part a Letk be the ephstaht ofpropomonallty S ls lnversely propomonal to x ls expressed by the equatror below k Part h To nd It subsumte x 9 and S 2 mto the formula S Secl39nn m Dilecl Variarpn Inverse Variarpmml Juianzr39n nn Page 5 an Example 5 P ls dlrectly propomonal to M a Wnte afotmulato express the glven statement b If M e s then P 24 Use the formula m pan a and the glven lnfol39matlon to fmdthe eonstant ofpropomonallty Snllltinn tn Example 5 Part a Let It be the eonstant of propomonallty P ls ddrecdy propomonal to M ls expressed by the equatton below PkM Part h To nd It subsatute M e s and P e 24 mto the formulaP W 24 k6 24 6k To solve for k dlvlde both sldes by 6 E 6 1 41t Example In a certam area the propenytax on ahouse ls dlrecdy propomonal to tts assessed value Ahouse thatts assessed at 125000 has apropenytax of 02950 Fmdthe propenytax on ahouse thathas an assessed value of 175000 Snllltinn tn Example 7 Secl39nn 3 Direct Varialhn lhverse Varialhnahd Juianzr39n nn Page a at 9 Let T he propertytax oh ahouse m 0 Let the assessed value of ahouse m 0 Letk be the eohstaht ofpropomonallty We are glyeh that the property tax or ahouse ls ddrecdy proporuohal to rts assessed value Thls ls expressed by the equatroh below T1tV Substltute T 2950 and V 125000 mto the formula 71tVto fmd 1t 2950 1t125000 2950 1250001t To solve for It dwlde both sldes by 125000 2950 Wk 125000 W 0 0236 It Now subsutute 1t 0 0236 mto the formula T kV T 0 0236V To nd the property tax or ahouse thathas ah assessed value 175000 subsutute y175000 rhto the formula T 0 023w T 0 0236175000 130 The property tax or a 175000 house ls 4130 Secl39nn m Dilecl Vzrizl39nnt htmse vmmhmm Juinlvzr39n nn Pig 7 ul 9 Examples R lsjolndy proportional to S and T a wttte afotmmato express the gtven statement b 4 and T s then R 21 Use the fotmtuattt pan a and the gtven tnfotmatton to nd the eonstsht of proportionality Snlntinn tn Example 8 Part a Letk be the eonstant ofpropomonahty R ts totnt1yptopomona1 to S and T ts expressed by the equation below R m Part h To nd It subsumte 524 T s andR 21 tnto the formula m 21 k46 21 24k To solve fork dwlde both states by 24 a 24k 24 24 Z 8 Example9 The eost offrammg apostettstotnuy proportional to tts 1ettgth and wtdth If apostetthat ts 3 feetlong and 2 feet wtde eosts 90 to tame whatts the east offrammg apostetthatts 4 feet by 2 5 feetv Snlntinn tn Example 9 Secl39nn 3 Direct Variarnn lnverse Varialhnana Juianzr39n nn Page a hf 9 Let c the eost of framlng aposter tn 3 LetL the length ofthe poster tn feet Let W the wldLh ofthe poster tn feet Letk be the eonstant ofproporuonahty We are glven that the eost offramlng a poster ls Jolntly propomonal to the length and wldLh Thls ls expressed by the equataon below CkLW Substltute c 90 L 3 and W 2 lnto the formula c kLW to nd k 90 k3 2 90 6k To solve for k dwlde both sldes by 6 a g 6 I 151c Now subsutute 1t 15lnto the formula c kLW C15LW To nd the eost offramlng aposterthat ls 4 feet by 2 5 feet subsutute L 4 and W 2 5 rnto the formula C15LW c l542 5 150 The eost offramlng aposter that ls 4 feet by 2 5 feet ls 150 Exercise Set 33 Variation Write a formula to express each of the following state m 1 2 10 11 12 ents P varies inversely as W y varies directly as T c is jointly proportional to h and r F is inversely proportional to M A is directly proportional to the square root of C d is jointly proportional to q and R H varies directly as Y and inversely as the square of J B is proportional to n and inversely proportional to the cube of q G is jointly proportional to x and z and inversely proportional to b and 0 Z varies directly as y and inversely as x and w Vis jointly proportional to l w and h Y varies jointly as F G andR and inversely as the square root of V Write a formula to express each statement Then find k the constant of proportionality 13 14 15 16 17 18 19 Tis proportional toM IfM 5 then T 30 B varies inversely as Q IfQ 5 thenB 10 J is jointly proportional to L and E If L 6 and E 8 thenJ 25 P varies directly as A and inversely as the square ofN IfA 2andN 9 thenP 7 v is proportional to w and inversely proportional to q and the square root of L If w 10 q 2 and I 25 then v 9 G is jointly proportional to the square roots of m n andpIfm1n9andp16thenG72 L varies jointly as B X and D and inversely as the cube root ofM IfB 3X 6 D 2 and M 8 then L 9 20 N is directly proportional to W and inversely proportional to A and T If W is three times as large as A and T 6 thenN 11 For each of the following problems a Write a formula to express the situation b Find the constant of proportionality c Solve for the indicated quantity 21 The cost of carpet is directly proportional to its area If a room measuring 10 ft by 12 ft costs 270 to carpet find the cost of carpeting for a room that measures 8 ft by 9 ft 22 The cost of a custommade Window is proportional to its area If a Window measuring 30 inches by 50 inches costs 500 find the cost of a Window that measures 11 inches by 25 inches 23 According to Hooke s Law the force needed to stretch a spring is proportional to the amount that the spring is stretched If a spring is stretched 10 inches by a force of 15 pounds how much will the spring stretch if a force of 195 pounds is applied 24 The revenue of a furniture store varies directly as the amount of money spent in advertising If the store spends 4000 in advertising one month and their monthly revenue is 510000 how much should they spend in advertising if they want next month s revenue to be 700000 25 At a factory the amount of time to make a given number of dresses varies directly as the number of dresses and inversely as the number of seamstresses working If it takes 4 hours for 5 seamstresses to make 12 dresses how long would it take 2 seamstresses to make 9 dresses 26 The intensity of an xray beam at any distance from the source varies inversely as the square of that distance If the intensity of an xray beam is 30 Rhour when the obj ect is 2 inches from the source determine the level of intensity when an object is 5 inches from the source Note Rhour stands for Roentgenshour A Roentgen is a unit for measuring the amount of xray or gamma radiation in the air

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