a We see that the bow of the canoe is tied to the dock at the point O. Since the length of the rope is 10 feet, the canoe can be at any distance less than 10 feet. When we stretch the rope and rotate it, the rope will scan a sector S_1 of a circle with a radius of 10 feet.
If we keep rotating, another sector S_2 of a circle with a radius 2 feet will be formed. This is because the width of the dock is 2 feet less than the length of the rope.
b Let's start by calculating the area of S_1. This sector formed by a 270 ^(∘) arc of a circle with radius of 10 feet. Therefore, the area of this sector is 270360 of the area of that circle. Let's keep this area in terms of π for now.
Next, we will calculate the area of S_2. Since S_2 is bounded by a 90^(∘) arc of a circle with radius 2 feet, the area of S_2 equals 90360 of the area of that circle. Let's calculate this area in terms of π as well.
The sum of the areas of S_1 and S_2 equals the area of the region in which the bow of the canoe can travel. Let's add them and calculate the final area.
