Problem: you need to make an open-top box from a rectangle of cardboard 9 by 13 inches by cutting a square from each corner and folding up the sides. you want the box to have the maximum possible volume. complete the following problems showing all your work: if each square you cut out is x by x inches, write an expression for the volume of the box, v(x), in terms of x. a sketch may . show all your work. sketch (or use technology to plot) the function to find the maximum. clearly label or state the (x, y) coordinates of the value of x that optimizes the function and the maximum volume (y). how many dollars worth of quarters (25 cent pieces) would it take to fill the box? show all your work!

The function is V(x) = 117x-44x²+4x³.
The screenshot of the graph is attached.
The maximum of the function is (1.7, 91.392), with the maximum volume being 91.392 in³.
$300 worth of quarters, laying side by side, would fill up the box.

Explanation
Let x be the amount cut from each corner.  Since all 4 corners will be cut, 2x will be cut from the length of 13 inches and 2x will be cut from the width of 9 inches.  This gives us a new length of (13-2x) and a new width of (9-2x).  The height of the box will be the x cut from each corner.  

This gives us a volume of 
V(x) = (13-2x)(9-2x)(x)

Multiplying the two binomials, we have
V(x) = (13*9-2x*13-2x*9-2x(-2x))(x)
V(x) = (117-26x-18x+4x²)(x)

Using the distributive property we have
V(x) = 117*x-26x*x-18x*x+4x²*x
V(x) = 117x-26x²-18x²+4x³

Combining like terms gives us
V(x) = 117x-44x²+4x³.

The graph is attached showing the maximum, (1.7, 91.392).

To find the number of quarters that will fit:
A standard U.S. quarter has a diameter of 0.955 in and a width of 0.069 in.

The new length of the box will be 13-2(1.7) = 13-3.4 = 9.6.  This means we can fit 9.6/0.955 = 10.05≈ 10 quarters across the length.

The new width of the box will be 9-2(1.7) = 9-3.4 = 5.6.  This means we can fit 5.6/0.955 = 5.86 ≈ 5 quarters across the width.  

This means each "layer" will hold 5(10) = 50 quarters.

The height of the box is 1.7 inches; this means we can fit 1.7/0.069=24.64≈24 layers of quarters.

This gives us a total number of quarters of 50(24) = 1200

This gives us 1200(0.25) = $300.