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. see the illustration. prove that the area of the opening is a function a(theta)

Answer 1

The area of the opening can be expressed as the function of theta given by the function: a(θ) = 16 sin θ ⋅ (cos θ + 1).

We are given that:

Width of the sheet = 12 inches.

Length from each end that is being marked = 4 inches.

Length is also bent on the angle θ.

So, we get the function of area in terms of theta as:

a(θ) = 16 sin θ ⋅ (cos θ + 1).

Therefore, the area of the opening can be expressed as the function of theta given by the function: a(θ) = 16 sin θ ⋅ (cos θ + 1).

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Answer 2

area of the opening 14.93 sq. inch.

From this statement "the area of the opening may be expressed as the function: a(θ) = 16 sin θ ⋅ (cos θ + 1).

if θ = 30°

We can substitute 30° in the expression a(θ) = 16 sin θ ⋅ (cos θ + 1) to get the area of the opening.

The area of the opening = 16 sin 30 ⋅ (cos 30 + 1)

                                       = 16x0.5 (0.866 + 1)

                                       = 8(1.866)

                                       = 14.93 sq. inch.

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