Calculus question help

Question

A box with an open top is to be constructed from a square piece of cardboard, 2 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

A rectangular plot of land is to be enclosed by a fence. Furthermore an additional section of fence is set parallel to one side, cutting the plot into two smaller rectangles. If 72 meters of fence are available, what is the greatest area of land that can be fenced in this manner?

Answer 1

let 2 feet width = 24 inches

the the sides of the box are

V = x(24-2x)^2 where x is the side of the cut out square of each corner

expand, take the derivative, set = 0 and solve for x

4x^3 -24 x^2 + 144

V' = 3x^2 - 48x + 144 =0

x^2 -16x +46 = 0

x = 8-4.2 = 3.8 inches = close to 1/3 foot

largest volume is about 3.8(24-7.6)^2 cubic inches

or roughly 1/3 ft by 4/3 ft by 4/3 ft = about 16/27 cubic feet