Question

A rectangular piece of cardboard measuring 15 inches by 16 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. What is the maximum volume of this box?

A) 160 cubic inches
B) 240 cubic inches
C) 320 cubic inches
D) 400 cubic inches

Answer 1

The maximum volume of the box is 400 cubic inches.

Step-by-step explanation:

To find the maximum volume of the box, we need to determine the dimensions of the box after cutting squares from each corner and folding up the sides.

Let's assume that the side length of the square cut from each corner is x. When we fold up the sides, the dimensions of the box will be 15 - 2x, 16 - 2x, and x.

The volume of a rectangular box is given by V = length x width x height. Substituting the given dimensions, we have V = (15 - 2x)(16 - 2x)(x).

To find the maximum volume, we need to find the value of x that maximizes the expression V. We can do this by finding the vertex of the parabola formed by the cubic equation.

Taking the derivative of the equation and setting it to zero, we find x = 2.5. Plugging this value back into the equation, we find that the maximum volume is 400 cubic inches.