1. A compact disc has the shape of a circle of diameter 5 inches with a 1-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity?
Solution: 2
Doubling the capacity is equivalent to doubling the area, which is initially $\pi\left[(5 / 2)^{2}-\right.$ $\left.(1 / 2)^{2}\right]=6 \pi$. Thus we want to achieve an area of $12 \pi$, so if the new diameter is $d$, we want $\pi\left[(d / 2)^{2}-(1 / 2)^{2}\right]=12 \pi \Rightarrow d=7$. Thus we need to add 2 inches to the diameter.