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Question

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 13 inches by 25 inches by cutting out equal squares of side (X) at each corner and then folding up the sides....Express the volume (V) of the box as a function of (X). Then use your graphing calculator to graph the function and find the (X) value that produces the maximum volume. Round your answer to the nearest hundredth

Answer 1

This is a lot easier to see if we could draw or upload a figure.   I will try below with dashes, but it will be rough...

 

 

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So each of those little short edges are supposed to be squares of side lengths x.

 

Since the long side of the rectangular cardboard is 25 inches long, if you take "x" inches out of each corner, the remaining part of the cardboard, the part that will be folded up to make a vertical edge of the box (the lengths) is

(25 - 2x) inches long.

 

The short side of the cardboard is 13 inches.  Taking "x" inches from both corners makes the remaining part of the cardboard along the short sides (the widths)  (13 - 2x)  inches wide.

 

The Volume a a rectangular box is  length * width * height.

 

The height is how tall the box will be, and that will be "x".

 

So the equation for the volume is

 

V = (25 - 2x) * (13 - 2x) * x

 

 

"FOIL" the first two, then distribute the "x" in, and you will get a cubic function that your teacher wants you to graph.

 

 

( I get that  V = (325 - 76x + 4x²)(x)   or

V = 325x - 76x² + 4x³

 

 

This is what your teacher wants you to graph.  

 

I found that the maximum volume  is at approximately x = 2.72