pg. 820, #47
Find the dimensions of a rectangular box
of maximum volume such that the sum of the lengths of its
12 edges is a constant c.
ans: cube, edge length c/12
Solution:
Suppose that the dimensions of the box are x, y, and z. We are given
that 4 x + 4y + 4z = c. Thus if
V is the volume of the box then V = xy(c/4 - x - y) or
.
Next,
By the geometry of the problem we must have
.
Thus the critical point equations
reduce to a set of two equations in two
unkowns, which is easy to solve:
The box of maximium volume is, then, a cube of side c/12.
