Question

alberta diploma question
a rain gutter is made from sheets of aluminum that are 32 cm wide. the first step in forming the rain gutter is to turn the edges up to form right angles, as shown in the diagram below
the cross - sectional area formed by the turned - up edges affects the water flow. this cross - sectional area, a, can be modelled by the function
$x(32 - 2x)$
where x is the height of the turned - up edges.
to the nearest tenth of a centimetre, the height of the turned - up edge, x, that will maximize the cross - sectional area is
cm

Explanation:

Step1: Expand the function

$$A(x)=x(32 - 2x)=32x-2x^{2}$$

Step2: Find the derivative

The derivative of \(A(x)\) using the power rule \((x^n)^\prime=nx^{n - 1}\) is \(A^\prime(x)=32-4x\).

Step3: Set the derivative equal to zero

Set \(A^\prime(x) = 0\), so \(32-4x=0\).
Solve for \(x\):
\(4x=32\), then \(x = 8\).

Answer:

\(8.0\)