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.)

Question

Mathematics

Rihanna Pham

17 November, 19:43

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.)

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Rapunzel · Answer 1

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.