Question

A rectangular gutter is formed by bending a 24 cm wide sheet of metal as shown. As a whole number, what is the height of a the gutter of maximum capacity?

Answer 1

Let $x=$the height of the gutter. Then the width of the gutter will be $24-2x.$

The capacity is

$$S=S(x)=x(24-2x), 0<x<12$$

The quadratic function has the absolute maximum at

$$x=-\dfrac{24}{2(-2)}=6$$

The height of a the gutter of maximum capacity is 6 cm.