Volume.A rectangular box with a square base is to be formed from a square piece of metal with 54-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?
A) 27
B) 2
C) 9
D) 14
E) 1

Question

​Volume.A rectangular box with a square base is to be formed from a square piece of metal with 54-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? ​  ​
A) ​27 
B) ​2 
C) ​9 
D) ​14 
E) ​1

Quizplus · Accepted Answer

Let x be the side of the square cut from each corner. Then the length of each side of the base of the box is (54-2x). The height of the box is x. The volume is given by:

V = (54-2x)^2 * x

Differentiating with respect to x and setting the derivative equal to 0, we get:

dV/dx = 2x(54-2x)^2 - 2(54-2x)(54-4x) = 0

Simplifying and solving for x, we get:

x = 9

To confirm that this value maximizes the volume, we can check the second derivative:

d^2V/dx^2 = -24(54-2x) < 0 for x = 9

Since the second derivative is negative at x = 9, this is a maximum point. Therefore, the value of x that maximizes the volume of the box is x = 9.

​VolumeA Rectangular Box with a Square Base Is to Be

VolumeA Rectangular Box with a Square Base Is to Be