The Dog's Span I

The maximum area that can be reached by the dog can be obtained by stretching its chain to its maximum distance from the point where it is tied. Since the chain is not elastic, then it will form a circular area around the point where it is tied (as is shown in the red region of the figure above).

The area is found using $\large A =\frac {\theta r^{2}}{2}$ . In this case, the angle $\large \theta$ is obtained by determining the exterior angles of the triangle. Since it is an equilateral triangle, then the angle is found to be $\frac {2}{3} \pi$ . Also, $r = 12$ in this case.

When the dog will try to reach another side of the pillar, its reach will decrease since 5m of its chain is restricted by the first side of the pillar. Thus, the circular area formed will have a radius $r = 7$ as is shown in the blue region of the figure above.

Can the dog still reach the third side of the pillar? Yes. Since the free chain is 7m long, it can still reach the third side and reach a circular area of $r = 2$ as is shown in the green region of the figure above.

The total area is then

$\large \large A_{total} = \frac {1}{3} \pi ( 12^{2} + 7^{2} + 2^{2} )$ $\large \large = \frac {197}{3} \pi$

$\large \large = 65 \frac {2}{3} \pi$

With this answer, we find that $a + b + c$ equals to $\large \boxed {70}$ .

I highly suggest you try this problem too.