a) Minimize S = x + 9y with xy = 9 and both x and y > 0 b)For a rectangle with a
ID: 3192307 • Letter: A
Question
a) Minimize S = x + 9y with xy = 9 and both x and y > 0 b)For a rectangle with a perimeter 56 to have the largest area, what dimensions should it have? (Enter the smaller value first.) c) Assume that it costs Apple approximately C(x) = 32,400 + 100x + 0.01x^2 dollars to manufacture x 30-gigabyte video iPods in a day. How many iPods should be manufactured in order to minimize average cost? and What is the resulting average cost of an iPod?Explanation / Answer
a). S = x + 9y ......xy =9 ==> y =9/x .....S = x+81/x ==> dS = 1-81/x^2 =0 ===> x^2 =81 ==> x = +/- 9 ..gives y = +/-1 but given x and y >0 ..so x=9 and y =1 and d^2S = 81/2x^3 = 1/18 >0 .....so S have a minimum here. ....at this point S = 9 + 9 *1 =18 .........................b) 2(x+y) = 56 ==> x+y=28 ==> y = 28-x ......area A = xy = x(28-x) ....==>dA = 28 - 2x = 0 ==> x =14 and y =14 and at this value d^2A = -2 dC(x) = 100 + 0.02 x = 0 ==> x = -5000 (meaningless) and the cost is minimum at this x because d^2C(x) = 0.02 >0 and it has a negative average cost again which is meaning less .............the avg cost can be minimized if x =0 i.e. they do not manufacture any iPods .....minimized avg cost = $32400........Related Questions
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