SOLUTION: Given a rectangular sheet of cardboard that is 15 inches by 25 inches, a small square of the same size is cut from each corner. Each side of the cardboard is folded along the cuts

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Question 237557: Given a rectangular sheet of cardboard that is 15 inches by 25 inches, a small square of the same size is cut from each corner. Each side of the cardboard is folded along the cuts to form a lidless box.
A)What is the maximum volume, V(x), of the box?
B)What size of the cut squares would produce a box with a volume equal to 400 cubic inches?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
Answer to B:

None.

Maximum Volume of the box is going to be less than 600 cubic inches.

The maximum volume of the box is going to be somewhere around 100.

Here's how I figured.

Let x = the length of one side of each square.

If you cut a square off each corner of the rectangle, then the length will be reduced by 2x and the width will be reduced by 2x.

The height of your box will be x.

the length of your box will be 25 - 2x.

The width of your box will be 15 - 2x.

The volume of your box is calculated as V = (25-2x) * (15-2x) * x

To find the rots of this equation, set it equal to 0 and solve.

You get:

(-2x+25) * (-2x+15) * x = 0

This equation would cross the x-axis at:

x = 0 and x = 12.5 and x = 7.5

A graph of this equation would look like:

graph%28400%2C400%2C-20%2C20%2C-600%2C600%2C%28-2x%2B25%29%2A%28-2x%2B15%29%2Ax%29

The domain of this equation has to be x > 0 to x < 7.5.

When x = 0, you have a flat sheet of cardboard with no height.

When x = 7.5, the width would be equal to 0 because 15 - 15 = 0.

This means that the range of this equation would be a minimum height > 0 and a maximum height < 7.5

The volume of the box would have to be be a minimum > 0 and a maximum of around 500 cubic inches as shown on the graph.

This would occur at somewhere around x = 3

Without looking at the graph, this would be difficult to solve.

It was only after looking at the graph that I was able to see what was going on.

There is a maximum point but I don't know the formula to find it given the equation as it stands.

It's not a quadratic and doesn't look like it can be made into a quadratic easily.

Best I can do.

Hope it helps just a little.