Question

A dog is leashed to the end of an infinitely long wall with angled ends in an infinitely large

grass field, with measurements as shown. If the area of the region of grass the dog can walk
on can be written with the expression a/bπ, what is a + b?

Answer 1

Let
`\theta` be the angle the dog makes with the positive horizontal. (So in the given sketch, we see the dog at an angle of
`\theta=\frac\pi4`, for example.)

When
`\theta` ranges from
`-\frac\pi2` to
`\pi`, so the angle subtended by the arc traced out by the dog in this range measures
`\frac{3\pi}4`, it can cover a total area of 3/4 of a circle with radius 5 m, or
`\frac{3\pi}4 (5\,\mathrm m)^2 = \frac{75\pi}4 \,\mathrm m^2`.

When
`\theta=-\frac\pi2` or
`\theta=\pi`, the leash is flush against the wall, at which point the remaining free space is the sector of a circle subtended by an angle of 180 - 135 = 45 degrees, or
`\frac\pi4` radians, with radius 4 m. There's one sector like this on either side of the wall, so the remaining area the dog can reach is
`2\cdot\frac\pi4(4\,\mathrm m)^2 = 8\pi\,\mathrm m^2`.

Then the total area the dog can cover is

`\frac{75\pi}4 + 8\pi = \frac{107\pi}4 \,\mathrm m^2`

so that
`a=107` and
`b=4`, and hence
`a+b=\boxed{111}`.