Let the side length of the square ends be x, and the height of the box be h.
The total length (sum of length, width, and height) of the box is x + x + h = 2x + h.
The girth (distance around) of the box is 2x + 2h = 2(x + h).
Given that the sum of the length and girth is at most 108 inches:
2x + h + 2(x + h) ≤ 108
2x + h + 2x + 2h ≤ 108
4x + 3h ≤ 108
h ≤ (108 - 4x) / 3
The volume of the box is V = x^2 * h.
Substitute the expression for h in terms of x into the volume formula:
V = x^2 * (108 - 4x) / 3
V = (108x^2 - 4x^3) / 3
To find the dimensions of the largest acceptable box, we need to find the maximum volume. We can do this by taking the derivative of the volume function and setting it equal to zero:
dV/dx = (216x - 12x^2) / 3
0 = 216x - 12x^2
0 = 12(18 - x)
x = 18
The maximum volume occurs when x = 18 inches.
Substitute x = 18 into the expression for h:
h = (108 - 4(18)) / 3
h = 24
Therefore, the dimensions of the largest acceptable box with square ends are 18 inches by 18 inches by 24 inches.
1.The U.S. Post Office will accept a box for shipment only if the sum of the length and girth
(distance around) is at most 108 inches. Find the dimensions of the largest acceptable box with
square ends.
1 answer