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 = 45°, what is the area of the opening?

Answer 1

`4(2+√(2))\text{ square unit}`

Explanation:

Given function that shows the area of the opening,

`A(\theta)=16 \sin\theta (\sin \theta + 1)`

If
`\theta = 45^(\circ)`

Hence, the area of the opening would be,

`A(45^(\circ))=16 \sin 45^(\circ) (\cos 45^(\circ) + 1)`

`=16* (1)/(√(2))* ((1)/(√(2))+1)`

`=16((1)/(2)+(1)/(√(2)))`

`=8+(16)/(√(2))`

`=8+4√(2)`

`=4(2+√(2))\text{ square unit}`