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 piece cut from each corner be x. Then the dimensions of the base of the rectangular box would be (30-2x) by (30-2x), and the height of the box would be x.

So the volume of the box would be:

V = (30-2x)^2 * x
V = 4x^3 - 120x^2 + 900x

To find the value of x that maximizes this volume, we need to take the derivative of V with respect to x and set it to zero:

dV/dx = 12x^2 - 240x + 900 = 0
x^2 - 20x + 75 = 0
(x-5)(x-15) = 0

So x can be either 5 or 15. To determine which one gives the maximum volume, we need to evaluate V at each value:

V(5) = 1250 cubic inches
V(15) = 3375 cubic inches

So the maximum volume is achieved when x = 15, and the answer is choice C.

A Rectangular Box with a Square Base Is to Be