Worksheet 24: Optimization

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1 Worksheet 4: Optimiztion Russell Buehler 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, P (I + I + 4)(100) 100I(I + 1) (I + I + 4) (100)(I + I + 4 I I) (I + I + 4) (100)( I + 4) (I + I + 4) (100)( I + )(I + ) (I + I + 4) Setting the top equl to 0, 0 ( I + )(I + ) I, Note further tht the derivtive is undefined whenever the bottom of the frction is zero but this only occurs when the bottom of the originl function is lso zero, nd so these points re not in the domin of the function. We need only compre I, : And thus I is the unique mximum. P () 0 P ( ) Find the point on the curve y x tht is closest to the point (3, 0). Note tht we wish to minimize the distnce between the given function nd the given point; we use, therefore, the distnce formul: d (x 3) + (y 0) Furthermore, since we only consider points on the function y x, d (x 3) + ( x 0) Noting tht d must be positive, nd so mximizing d is the sme s mximizing d, we consider Tking the derivtive, d (x 3) + ( x 0) (x 3) + x d dx [d ] (x 3) + 1 x 5

2 Setting the derivtive equl to zero, x 5 0 x 5 5 y 3. Find the re of the lrgest rectngle tht cn be inscribed in right tringle with legs of lengths 3 cm nd 4 cm if two sides of the rectngle lie long the legs. Note first tht the formul we would like to mximize is A (4 x)(y). It remins, then, to eliminte either x or y from the eqution (we need x in terms of y or vice vers). To do so, note tht the lrge tringle bove is similr to the tringle bove the rectngle (they re both right nd shre n ngle, nd thus hve the sme ngle mesures for ll ngles). We therefore hve, x y 4 3 4y 3x y 3 4 x Substituting into our re eqution, A (4 x)( 3 4 x) 3x 3 4 x Tking the derivtive, A 3 3 x nd setting it equl to 0, And so the requested re is x 0 x

3 4. ( ) Find the re of the lrgest rectngle tht cn be inscribed in the ellipse x + y b 1. The eqution we wish to mximize is A (w)(z). Note first tht the ellipse bove hs width long the x-xis nd height b long the y-xis. To eliminte either w or z, we solve the eqution of the ellipse for its positive component: x + y b 1 y (b )(1 x ) b b x Note next tht ny cceptble w vlue will be equl to y in the eqution bove for some x. Similrly, note tht the x vlue will be the sme s z! We hve, then, w b b z And so, substituting into the re eqution, Tking the derivtive, A 4z A 4 b b z b b z + (4z) b z b b z 4(b b z ) + 4b z b b z b b z 4(b b z ) 4b b b z b 8b z b b z z nd setting it equl to 0, b 8b z 0 z 8

4 Noting tht mking the bottom of the derivtive equl to 0 is the sme s mking w 0 nd is thus not mximum, the requested re is: A 4( ) b b ( ) 4( 7 ) 8 b 7b 5. A cone-shped drinking cup is to hold 7cm 3 of wter. Find the height nd rdius of the cup tht will use tht smllest mount of pper. Noting tht the surfce re of cone is given by We wish to minimize: SA πr + πrl P πrl Furthermore, we re given tht the volume of the cone must be 7cm 3 : V πr h 7 h πr Anlyzing the digrm bove, note tht by the Pythgoren theorem, l h + r, nd so: l ( πr ) + r Substituting into P, Tking the derivtive, P πr ( πr ) + r πr π r 4 + r P π π r 4 + r + (πr) π( π r + r ) 4 + π r + r 4 πr 4 ( 4) π r 5 + r π r 4 + r ( ) π r 4 + πr π r 4 + r + πr + ( ) π r 4 + πr π r 4 + r π r 4 + πr π r + r 4

5 Setting the top equl to 0, π r 4 + πr 0 πr 4 + π r 6 πr 4 nd so, + π r r 6 π h π( π ) 3 4 π( π ) 34 3 π π Note tht no undefined vlues for the derivtive re cceptble since they give l 0 nd r 0 respectively. 6. At which points on the curve y x 3 3x 5 does the tngent line hve the lrgest slope? We wish to mximize y 10x 15x 4. Tking the derivtive, y 40x 60x 3 Setting it equl to zero, 40x 60x 3 0 x(40 60x ) x 0 x 4 And thus, x 0, ± Compring these points, f (0) 0 f ( ) f() 40 The points which mximizes the tngent line slope re therefore (, 5) nd (, 3). 7. () Show tht if the profit P (x) is mximum, then the mrginl revenue equls the mrginl cost. Assuming both mrginl revenue (MR) nd mrginl cost (MC) exist, these re interpreted s the derivtive of revenue (R) nd cost (C) respectively. The profit function is P (x) R C, nd so ssuming no undefined derivtive vlues P (x) 0 when MC or C is equl to MR or R the mximum by the extreme vlue theorem (ssuming, gin, tht the derivtive is defined on the entirety of the originl domin).

6 (b) If c(x) 16, x 1.6x +.004x 3 is the cost function nd p(x) x is the demnd function, find the production level tht will mximize profit. Note tht the revenue function is given by xp(x) tht is, the price p(x) times the number of items x. Thus, R(x) 1700x 7x Finding MC nd MR, MR x MC x +.01x Setting them equl, x x +.01x x +.01x x x x + 1x 1(x 100)(x ) And thus, x 100

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