Question

The base of a circular fence with radius 10 m is given by x = 10 cos(t), y = 10 sin(t). The height of the fence at position (x, y) is given by the function h(x, y) = 5 + 0.05(x2 − y2), so the height varies from 0 m to 10 m. Suppose that 1 L of paint covers 100 m2. Determine how much paint you will need if you paint both sides of the fence. (Round your answer to two decimal places.)

Answer 1

Answer:

  6.28 L

Step-by-step explanation:

The length of the circular fence is ...

  C = 2πr = 2π(10 m) = 20π m

The height in terms of t is ...

  h(t) = 5 +0.05((10cos(t))^2 -(10sin(t))^2) = 5 +5(1 -2sin(t)^2)

For 0 ≤ t ≤ 2π, the average value of this height function is 5.

The fence has an average height of 5 m and a length of 20π m, so an area of ...

  (one side) fence area = (5 m)(20π m) = 100π m²

The area of both sides is double this, or ...

  total fence area = 2(100π m²) = 2π(100 m²)

Since 1 liter covers 100 m², we need 2π liters to cover the fence.

The amount of paint needed is 6.28 L.