2 calculus questions

[匿名]

2011-11-26 8:08 pm

1. A parcel delivery service will deliver a package only if the lengthplus the girth (distance around, taken perpendicular to the length)does not exceed 104 inches. Find the maximum volume of a rectangularbox with square ends that satisfies the delivery company'srequirements.

2. For the cost function
圖片參考：http://r140.n100.queensu.ca:8081/wwtmp/equations/57/8e693aa5480b10e51620f2605fefab1.png
find

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(a) The production level that will minimize the average cost.

Ans: 62.996

(b) The minimal average cost

Ans: 53.811

回答 (1)

用戶25975

2011-11-26 9:29 pm

✔ 最佳答案

1) Let x inches be a side of the square end, then we have:

Length L <= 104 - 4x

So taking max. possible volume, we let L = 104 - 4x, then vol. is:

V = x2(104 - 4x)

= 104x2 - 4x3

dV/dx = 208x - 12x2

d2V/dx2 = 208 - 24x

When dV/dx = 0,

208x - 12x2 = 0

4x(52 - 3x) = 0

x = 0 (rej.) or 52/3

When x = 52/3, d2V/dx2 = 208 - 24 x 52/3 < 0

Hence x = 52/3 gives max. V = 10415 cubic inches

2a) The avg. cost is:

A = C/x = 1000/x + 30 + 0.002x2

dA/dx = -1000/x2 + 0.004x

d2A/dx2 = 2000/x3 + 0.004

For dA/dx = 0:

1000/x2 = 0.004x

x3 = 250000

x = 62.996

Sub into d2A/dx2 = -0.004 > 0

Hence it gives a min. average cost

b) Min. avg cost = 53.811 when sub x = 62.996 into C/x

2011-11-26 13:30:50 補充：
For 2a) d^2A/dx^2 = 0.012 > 0 when x = 62.996

參考: 原創答案

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