Question

A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, this length is bent up at an angle θ. The area A of the opening may be expressed as the function: A(θ) = 16 sin θ ⋅ (cos θ + 1). If θ = 90°, what is the area of the opening?

Answer 1

we know that

The area A of the opening may be expressed as the function:

`A(\alpha) = 16 sin \alpha* (cos \alpha + 1)`

For
`\alpha =90`°

We simply have to substitute the value of the angle into the function

so

`A(90) = 16*sin 90* (cos 90 + 1)`

`A(90) = 16*(1)* (0 + 1)`

`A(90) = 16 in^(2)`

therefore

the answer is

the area of the opening is
`16 in^(2)`

Answer 2

We are already given with the function to solve for the area:
A(θ) = 16 sin θ ⋅ (cos θ + 1)

We simply have to substitute the value of the angle into the function. So,
If θ = 90°,
A(90°) = 16 sin (90°) ( cos (90°) + 1 )

Using the calculator or the definition of trigonometric functions at angle of 90°, we get the value of the area:
A(90°) = 16 square inches