An internet sales company will ship a box for free if the sum of the
length, width and height is 48 inches or less. What is the maximum
volume of a box 24 inches in length that ships for free ?
To find the maximum volume of a box that ships for free, we need to determine the maximum dimensions (length, width, and height) that would result in a sum of 48 inches or less.
Let's assume the width and height of the box are equal and denoted by x inches.
Given:
Length = 24 inches
Width = x inches
Height = x inches
According to the condition, the sum of the length, width, and height should be 48 inches or less:
24 + x + x ≤ 48
24 + 2x ≤ 48
2x ≤ 48 - 24
2x ≤ 24
x ≤ 12
Therefore, the width and height, or each side of the square base of the box, should be equal to or less than 12 inches.
The volume of a rectangular box is given by V = length * width * height. In this case, it becomes:
V = 24 * x * x
V = 24x²
To find the maximum volume, we need to maximize the value of x. Since x ≤ 12, plugging in the maximum value of x:
V = 24 * 12²
V = 24 * 144
V = 3,456 cubic inches
Therefore, the maximum volume of a box with a length of 24 inches that still qualifies for free shipping is 3,456 cubic inches.