Question

An open box is made from a square piece of cardboard 20 inches on a side by cutting identical squares from the corners and turning up the sides.(a) Express the volume of the box, V , as a function of the length of the side of the square cut from each corner, x. (b) Find and interpret V (1),V (2),V (3),V (4), and V (5). What is happening to the volume of the box as the length of the side of the square cut increases? (c) Find the domain of V.

Answer 1

v = x(20-2x)^2

using what you know about the shape of cubic curves, clearly there is a maximum volume for 0 <= x <= 10 (this is also the domain of v)

Naturally, volume is zero when x=0 and when x=10.

A little checking of values will show that v(x) is max when x = 10/3.