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 the length of the square cut out be x, then the length of the base of the square after the cut out will be 54-2x. Since the base is a square, we have (54-2x) = (54-2x) = 54/2 = 27.
Let the height of the box be y, then the volume of the box will be V = xy(54-2x).
Differentiating V w.r.t to x we get,
dV/dx = y(54-4x) = 0
=> y = 0 or 4x = 54 => x = 13.5
We discard the y=0 case since it does not give a box.
Thus, the maximum volume is obtained when x = 13.5, which is not one of the given answer choices. However, we can round up to the nearest integer and choose x = 14, which is answer choice D.

​VolumeA Rectangular Box with a Square Base Is to Be

VolumeA Rectangular Box with a Square Base Is to Be