Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Find the answer Italy

Problem:

A goat (indicated in the figure by $\Pi^{\alpha}$ ) is tied to a point $A$ with a rope 6 meters long as shown in the figure (seen from above). The segment $AB$ represents a fence 4 meters long beyond which the goat cannot jump. The square represents a building with side 2 meters inside which the goat cannot enter (and onto which it cannot climb). Determine the largest area (in $\mathrm{m}^{2}$ ) that the goat can graze.

Pick one

Solution

Solution:

The answer is $\mathbf{(A)}$ . Indeed, simple observations allow us to see that the area that can be grazed by the goat is the one shown in the following figure:

The area $S_{1}$ is a quarter of that of a circle of radius 6, that is: $\frac{1}{4}\left(6^{2}\right) \pi=9 \pi$ .

The area $S_{2}$ is a quarter of that of a circle of radius 4, that is: $\frac{1}{4}\left(4^{2}\right) \pi=4 \pi$ .

The area $S_{3}$ is a quarter of that of a circle of radius 2, that is: $\frac{1}{4}\left(2^{2}\right) \pi=\pi$ .

The area $S_{4}$ is half of that of a circle of radius 2, that is: $\frac{1}{2}\left(2^{2}\right) \pi=2 \pi$ .

The area sought is therefore: $S_{1}+S_{2}+S_{3}+S_{4}=16 \pi$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.