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Painting a Carousel

I've been listening to Car Talk for a few years now. The hosts Tom and Ray kept me company on my 2,000 mile trip from Iowa to California. Their mix of sarcasm and loving judgment reminds me of the times I’ve rooted around to fix things. Every week they post a puzzler. The problem statement for this week asks listeners to determine the area between two concentric circles given the length of a chord tangent to the inner circle.

Since I'm an engineer, I had to sketch it out:

Where
D, diameter of the outer circle

R, radius of the outer circle

d, diameter of the inner circle

r, radius of the inner circle

C, length of chord tangent to the inner circle

Given
C = 70 ft

Examining Figure 1, we see that r intersects C at its midpoint. Therefore,makes up one leg of a right triangle with r and R. The Pythagorean Theorem relates the three sides of a right triangle as

Substituting in our variables yields

Which can be rearranged as

Now we have one equation but two unknown variables. Let’s establish some relationships for area. The generalized equation for the area of a circle is

In terms of the outer circle

In terms of the inner circle

To solve the problem, we must find the area between the circles:

Recall from our Pythagorean relationship that we (rather fortunately) know the value of (R^{2}-r^{2}).

Ed, Biff, and Skip need enough paint to cover **3,846.5 square feet**.