Tom Sawyer’s aunt has asked him to whitewash both sides of an old fence. The fence line on the ground is in the shape of C(t) =
(30 cos3 t, 30 sin3 t) with 0 ≤ t ≤ π. The fence reaches up to the function f(x, y) = 1 + (y/3). Tom estimates that for each 25 square feet of whitewashing he lets someone do for him, the willing victim will pay 5 cents. How much can Tom hope to earn, assuming his aunt will provide the whitewash free of charge? You may find it helpful to see what this curve, a type of hypocycloid known as an astroid, looks like. A picture of an astroid can be found on page 127 in the text and Wikipedia has a sweet animation of how this curve is generated.
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