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.)

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.