Question

1) An airline requires that the total outside dimensions (length + width + height) of a checked bag not exceed 56 inches, Suppose you want to check a bag whose height equals its width. What is the largest volume bag of this shape that you can check on a flight? a) Length (in)? b) Width(in)? c) Height (in)? 2) Assume that It costs Apple approximately C(x) = 16,900 + 100x + 0.01x^2? dollars to manufacture x 32G8 iPods in a day.+ How many iPods should be manufactured in order to minimize average cost? Pods per day What is the resulting average cost of an iPod? (Give your answer to the nearest dollar.) HINT (see Example 1.) $ [ ] 3) A postal service accept packages only If the length plus girth is no more than 126 inches. (See the figure.) Assuming 5km that the front face of the package (as shown in the figure)is square, what is the largest volume package that the postal service will accept?

Answer 1

l + w + h <= 56

h = w

l + 2h <= 56

For largest volume, l + 2h = 56 --> L = 56 - 2h

V = lwh

V = (56 - 2h)(h)(h)

V = -2h^3 + 56h^2

dV/dh = -6h^2 + 112h = 0

-6h^2 + 112h = 0

h = 112/6

So, h = 56/3 inches

Therefore, w = 56/3 inches

l = 56 - 2h

l = 56 - 2(56/3)

l = 56 - 112/3

l = 56/3 inches

So, the dimensions are :
Length = 56/3 inches
Width = 56/3 inches
Height = 56/3 inches

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C(x) = 16900 + 100x + 0.01x^2

So, average cost = Ca(x) = (16900 + 100x + 0.01x^2) / x

Ca(x) = 16900/x + 100 + 0.01x

dCa/dx = -16900/x^2 + 0.01 = 0

16900/x^2 = 0.01

x^2 = 16900/0.01

x^2 = 1690000

x = 1300 ----> ANSWER

Average cost = 16900/1300 + 100 + 0.01(1300)

Resulting average cost = $126 ---> ANSWER

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3)

L + 2W + 2H <= 126

L + 2W + 2H = 126

If front face is square, then W = H

L + 4W = 126

L = 126 - 4W

Volume V = LWH

V = (126 - 4w)(w)(w)

V = -4w^3 + 126w^2

dV/dw = -12w^2 + 252w = 0

w = 252/12 ---> w = 21

So, h = 21

L = 126 - 4w --> 126 - 4(21)

L = 42

So, max volume is : LWH --> 42*21*21

V(max) = 18522 in^3 -----> ANSWER