Question

The base of a circular fence with radius 10 m is given by x = 10 cost, y = 10 sin t. the height of the fence at position (x, y) is given by the function h(x, y) = 4 + 0.01(x 2 − y 2 ). supposing that 1 l of paint will cover 10 square meters, determine the amount of paint needed to cover both sides of the fence.

Answer 1

The parametric equations for x and y describe a circle of radius 10 m, so the length of the base of the fence is the length of the circumference of a circle of radius 10 m. The formula for that circumference (C) is ...

... C = 2πr

... C = 2π·(10 m) = 20π m

The height as a function of angle (t) is found by substituting for x and y.

... h(t) = h(x(t), y(t)) = 4 + 0.01·((10cos(t))²-)10sin(t))²) = 4+cos(2t)

The average value of this over the range 0 ≤ t ≤ 2π is 4, since the cosine function has two full cycles in that range, and its average value over a cycle is zero.

Thus, the area of one side of the fence is that of a rectangle 20π m long and 4 m wide. That will be

... (20π m)·(4 m) = 80π m²

The amount of paint required to cover both sides of the fence is

... 2×(80π m²)×(1 L)/(10 m²) = 16π L ≈ 50.3 L

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You can work out the integral for area as a function of t. When you do, you will find it gives this same result.