Solve the problem.
-A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 102 in. What dimensions will give a box with a square end the largest
Possible volume?
A) 17 in. × 34 in. × 34 in.
B) 34 in. × 34 in. × 34 in.
C) 17 in. × 17 in. × 34 in.
D) 17 in. × 17 in. × 85 in.

Question

Solve the problem.
-A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 102 in. What dimensions will give a box with a square end the largest
Possible volume? 
A) 17 in. × 34 in. × 34 in. 
B) 34 in. × 34 in. × 34 in. 
C) 17 in. × 17 in. × 34 in. 
D) 17 in. × 17 in. × 85 in.

Quizplus · Accepted Answer

Let the dimensions of the square end be x by x, and let the length be y. Then the girth is 2x + 2y. We want to maximize the volume of the box, which is x^2y. The constraint is that 2x + 2y ≤ 102, or y ≤ 51 - x. We can substitute this inequality into the volume equation to get V(x) = x^2(51 - x). Taking the derivative and setting it equal to zero, we get 2x(51 - 2x) = 0, or x = 25.5. Since the dimensions must be integers, we round down to x = 25. Then y = 26, and the dimensions of the box are 17 in. × 17 in. × 34 in., which is choice C. Let x be the length of one side of the square end of the box, and y be the height of the box. Then the sum of the length and girth is given by x + 4y, and we want to maximize the volume V = x^2y subject to the constraint x + 4y ≤ 102. Using the method of Lagrange multipliers, we can set up the system of equations:

2xy = λ
x^2 = 4λ

Solving for λ gives λ = 2x^2/3. Substituting into the constraint, we get x + 4y = 2x^2/3, which simplifies to y = (2/3)x - x/12. To maximize V, we want to choose x to maximize y, since this increases the height and therefore the volume. Taking the derivative of y with respect to x, we get dy/dx = 2/3 - 1/12 = 5/12. Setting dy/dx = 0, we find x = 17.5, which gives y = 8.75 and V = 17.5^2(8.75) = 2703.13. Since this is greater than the volumes of the other choices, the best choice is C.

Solve the Problem. -A Private Shipping Company Will Accept a Box for Domestic