Question

A 22-inch by 70-inch piece of cardboard is used to make an open-top box by removing a square from each corner of the cardboard and folding up the flaps on each side. What size square should be cut from each corner to get a box with the maximum volume? Enter the area of the square and do not include any units in your answer.

Answer 1

Answer:

  25

Step-by-step explanation:

Let x represent the side length of the square in inches. Then the volume of the box is ...

  V = x(22 -2x)(70-2x) = 4(x)(x -11)(x -35)

  V = 4(x³ -46x² +385x)

The area will be maximized when the derivative of volume with respect to x is zero:

  dV/dx = 0 = 4(3x² -92x +385)

  0 = (3x -77)(x -5) . . . . . divide by 4 and factor

  x = 77/3 or 5

The appropriate domain of the volume function is x < 11, so the 77/3 answer is extraneous. The side length of the square is 5 inches, so its area is ...

  square area = (5 in)² = 25 in²

_____

It's an exercise left to the student to figure out how to enter that answer without the units.

_____

Comment on answers without units

The area can also be described as 161.29 square centimeters. Of course, 161.29 is very different from 25, so it is important to know what the units are. We appreciate that the computer checking the answer probably doesn't deal well with non-numeric answers, so the instructions are appropriate for someone actually feeding the computer. That is not a task I can help you with, so I don't need (and don't want) those instructions. In the real world a number without units is meaningless.