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

The correct answer is (A) $\pi$ P^2.

• Given Data and Principle: Two cows are tied to a common pole. The length of the rope for the first cow is P and for the second cow is Q, with the condition that P > Q. Each cow can graze in a circular region around the common pole with a radius equal to the length of its respective rope.
• Step-by-Step Derivation: The area grazed by the first cow with rope length P is $\pi P^2$$\pi$ P^2. The area grazed by the second cow with rope length Q is $\pi Q^2$$\pi$ Q^2. Since both cows are tied to the same common pole, the region grazed by the cow with the shorter rope (length Q) lies completely inside the larger circular region grazed by the cow with the longer rope (length P). Therefore, the total unique area grazed by both cows combined is simply the area of the larger circle.
• Verification and Option Contrast: The total grazed area is determined entirely by the maximum reach of the cows from the central pole, which is P. Thus, the total area is $\pi P^2$$\pi$ P^2. Options (B) and (C) do not represent the total combined unique space, and Option (D) incorrectly subtracts the inner area. Hence, Option (A) is correct.