Folded boxes Folded boxes Squares with sides of length x are cut out of each cor

Question

Folded boxes

Folded boxes Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. Suppose that in part (a) the original piece of cardboard is a square with sides of length l. Find the volume of the largest box that can be formed in this way. Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length l and L. Holding l fixed, find the size of the corner squares x that maximizes the volume of the box as L rightarrow infinity. (Source: Mathematics Teacher, November 2002) Making silos A grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. The metal in the dome costs 1.5 times as mu Ch as the concrete (per unit of

Explanation / Answer

Volume of a box=length*width*height

We can easily see that the height of the box will be x.

What about the length and the width?

The length should be 4 inches minus x inches minus x inches, since there would be two x's cut out of the length.

The width should be 3 inches minus x inches minus x inches, since there would be two x's cut out of the width as well. height=x height=x
length=4-2x

width=3-2x
volume=x*(4-2x)*(3-2x) inches cubed

(4x-2x^2) (3-2x) inches cubed (3-2x) inches cubed
You can multiply those terms out using the FOIL method and you will end up with the correct answer.

In some cases, it might be easier to leave the Volume in that first factored form.

I hope this helps!

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