Discussion
If two cows are tied to a common pole and the length of the ropes are P and Q, where P is greater than Q, then the total area grazed by both the cows will be ________ .
Answer: A
The total area grazed is \(\pi P^{2}\).
Both cows are tied to the same pole, so each grazes a circle centred on that pole: one of radius P and one of radius Q. Since P is greater than Q, the smaller circle lies entirely inside the larger one, and the total area grazed is just the area of the larger circle, \(\pi P^{2}\).
\(\pi P^{2} + \pi Q^{2}\) counts the inner circle twice. \(\pi P^{2} - \pi Q^{2}\) is the ring grazed only by the cow with the longer rope, and \(\pi Q^{2}\) is the area the smaller cow alone can reach.
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