a. The U. S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume? b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a).

Question

Mathematics

Jaylin Yoder

31 July, 13:36

a. The U. S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume? b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a).

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Answers (1)

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Trooper · Answer 1

The dimensions are 18 * 18 * 36in

Step-by-step explanation:

Let the lengths of the square ends be x in and the length be y in.

Then the maximum length + girth

= 4x + y = 108in

The volume V = x^2y.

From the first equation:

y = 108 - 4x

so V = x^2 (108 - 4x)

= 108x^2 - 4x^3

Finding the derivative:

V' = 216x - 12x^2 = 0 (for a maximum volume).

12x (18 - x) = 0

x = 18

So the value for x for a maximum volume = 18 in

This gives y = 108 - 4 (18) = 36 in.