A rectangular box with a square base is to be formed from a square piece of metal with 30-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is . What value of x will maximize the volume of the box?

Question

A rectangular box with a square base is to be formed from a square piece of metal with 30-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is  . What value of x will maximize the volume of the box? ​

Quizplus · Accepted Answer

Let the length of the side of the square cut from each corner be x.
Then, the dimensions of the base of the box will be (30-2x) by (30-2x).
The height of the box will be x.
Therefore, the volume of the box will be V = (30-2x)^2x.
To maximize V, we take the derivative of V with respect to x and set it equal to 0:
dV/dx = 4x(30-2x)^2 - 2(30-2x)^3 = 0
Simplifying, we get:
x(15-x)^2 = 45^2
x(225 - 30x + x^2) = 2025
x^3 - 30x^2 + 225x - 2025 = 0
(x-9)(x-15)(x-15) = 0
x = 9 or 15
Since x must be less than 15 (otherwise, there will be no metal left to fold up), the only valid solution is x = 9. Therefore, Choice C is the answer.

A Rectangular Box with a Square Base Is to Be