Questions: A piece of cardboard measures 80 inches by 120 inches. Evan wants to cut out 4 square corners of equal size to make a box with an open top. What size corners should be cut out to maximize the volume? What is the maximum volume? What are the dimensions of the box? Round your answers to two decimal places.

A piece of cardboard measures 80 inches by 120 inches. Evan wants to cut out 4 square corners of equal size to make a box with an open top. What size corners should be cut out to maximize the volume? What is the maximum volume? What are the dimensions of the box? Round your answers to two decimal places.
Transcript text: A piece of cardboard measures 80 inches by 120 inches. Evan wants to cut out 4 square corners of equal size to make a box with an open top. What size corners should be cut out to maximize the volume? What is the maximum volume? What are the dimensions of the box? Round your answers to two decimal places.
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Solution

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Solution Steps

Step 1: Establish the volume equation

The volume of the box is given by the product of its length, width, and height. The height of the box will be equal to the side length of the square cutouts, _x_. The length and width of the box will be the original dimensions minus twice the side length of the cutouts. Therefore, the volume _V_ is given by: _V_ = _x_(120 - 2_x_)(80 - 2_x_)

Step 2: Find the maximum volume

To maximize the volume, we need to find the value of _x_ that corresponds to the maximum value of _V_. This can be done by graphing the volume equation and finding the vertex, or by using calculus to find the critical points. The problem states to round to two decimal places, implying a graphing approach.

Graphing the equation _V_ = _x_(120 - 2_x_)(80 - 2_x_) shows a maximum volume of 67,603.58 cubic inches when _x_ is approximately 15.69 inches.

Step 3: Calculate the dimensions of the box

The height of the box is equal to _x_, which is approximately 15.69 inches.

The length of the box is 120 - 2_x_ = 120 - 2(15.69) = 88.62 inches.

The width of the box is 80 - 2_x_ = 80 - 2(15.69) = 48.62 inches.

Final Answer:

The size of the corners that should be cut out to maximize the volume is 15.69 inches. The maximum volume is 67,603.58 cubic inches. The dimensions of the box are approximately 88.62 inches by 48.62 inches by 15.69 inches.

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