a rectangular sheet of cardboard size 5' by 8' is to be used to make an open box by cutting out the four corners. What is the largest volume and be sure to explain how you found the MAXIMUM volume please and Thank You!

Answer 1

Are you allowed to use differential calculus?

Write an expression for box volume V in terms of the length of the corner cutouts, x.

V = (5-2x)(8-2x)*x
= 4x^3 -26x^2 +40x

Solve for x where V'(x) = dV/dx = 0

dV/dx = 12 x^2 -52x + 40 = 0
3x^2 -13x +10 = 0
x = (1/6)[13 +/- 7] = 1 and 3.333
The second root is not possible, because the removed material would exceed the length of the 5' side.

The largest volume is what you get with x = 1 foot.

Vmax = 18 ft^3