Question 2.6.8: A long rectangular sheet of metal, 12 inches wide, is to be ......
A long rectangular sheet of metal, 12 inches wide, is to be made into a r gutter by turning up two sides so that they are perpendicular to the sheet. many inches should be turned up to give the gutter its greatest capacity?
A long rectangular sheet of metal
12 inches wide
Two sides need to be turned up
The turned up sides should be perpendicular to the sheet.
Final Answer
The gutter is illustrated in Figure 9. If x denotes the number of inches turned up on each side, the width of the base of the gutter is 12-2 x inches. The capacity will be greatest when the cross-sectional area of the rectangle with sides of lengths x and 12-2 x has its greatest value. Letting f(x) denote this area, we have
\begin{aligned}f(x) & =x(12-2 x) \\& =12 x-2 x^2 \\& =-2 x^2+12 x,\end{aligned}which has the form f(x)=ax²+bx+c with a=-2, b=12, and c=0. Since f is a quadratic function and a=-2<0, it follows from the preceding theorem that the maximum value of f occurs at
x=-\frac{b}{2 a}=-\frac{12}{2(-2)}=3Thus, 3 inches should be turned up on each side to achieve maximum capacity. As an alternative solution, we may note that the graph of the function f(x)=x(12-2 x) has x-intercepts at x=0 and x=6. Hence, the average of the intercepts,
x=\frac{0+6}{2}=3 \text {, }is the x-coordinate of the vertex of the parabola and the value that yields the maximum capacity.