algebra
posted by lexis green
Amy's project in metal shop class is to make a box (with no lid). Her shop teacher gave her a piece of metal that is 28 cm long and 20 cm wide. The assignment is to form the box by cutting squares out of each corner and bending the sides as shown below. The shop teacher, Mr. Rust, will give extra credit to any student who forms the box that holds the most when the cutout square is a whole number of centimeters on each side. Amy wants the extra point! How large a square should she cut from each corner to form the box with the greatest volume?

stranger
Let x = side of square to cut off from each corner
20  2x = width of bottom rectangle
28  2x = length of bottom rectangle
V = x(20  2x)(28  2x) = 4x(10  x)(14 x) =
V = 4x(140  24x + x²)
V = 4(140x  24x² + x³)
Differentiating and equating to zero
dV/dx = 4(140  48x + 3x²) = 0
3x²  48x + 140 = 0
x = {48 ± √[48²  4(3)(140)] } /2(3)
x = {48 ± √624] } /6
x = {48 ± 4√39] } /6
x = (24 ± 2√39) / 3
x = (24 + 2√39) / 3 = 12.16 Discard, > ½20
x = (24  2√39) / 3 = 3.8 inch
Use x = 4 since whole number of cm is required.
So each square to cut is 4 cm on the side.
4 cm squares gives a box of 4 x 12 x 20 = 960 cm^3
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