Question

Rolly wishes to secure his dog with an $8$ -foot rope to a square shed that is $16$ feet on each side. His preliminary drawings are shown.

Which of these arrangements gives the dog the greater area to roam, and by how many square feet?

I, by $8\pi$

I, by $6\pi$

II, by $4\pi$

II, by $8\pi$

II, by $10\pi$

Solution:

In arrangement I the dog is tied at the middle of a side and sweeps a half-disk of radius $8$ : area $\frac12 \pi \cdot 8^2 = 32\pi.$ The rope reaches exactly to the corners, so nothing wraps.

In arrangement II the dog is tied $4$ feet from a corner. It sweeps the same $32\pi$ half-disk, and after the rope reaches the corner, $4$ feet remain to sweep a quarter-disk of radius $4$ : $\frac14 \pi \cdot 4^2 = 4\pi.$

So II gives $36\pi,$ exceeding I by $4\pi.$

Thus, the correct answer is C.

2006 AMC 10A Problem 12

Solution:

Problem 12 in Other Years