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.
