1. **Problem statement:** We have a circular island centered at $P(0,0)$ with radius 8 km. Four pirates want to divide the island into four equal areas by placing their bases so that the boundaries are the perpendicular bisectors between their bases. 2. **Step (a): Find coordinates of the four pirates' bases for equal area cells.** - Since the island is a circle centered at the origin, dividing it into four equal areas means dividing it into four equal sectors of 90° each. - The bases should be placed on the circle at points equally spaced by 90°. - Coordinates on a circle of radius 8 km at angles $0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$ are: - Pirate 1: $(8,0)$ - Pirate 2: $(0,8)$ - Pirate 3: $(-8,0)$ - Pirate 4: $(0,-8)$ 3. **Step (b): Calculate the area of each pirate's cell.** - The total area of the island is $A = \pi r^2 = \pi \times 8^2 = 64\pi$ km$^2$. - Dividing into 4 equal parts, each cell area is: $$\frac{64\pi}{4} = 16\pi \approx 50.3 \text{ km}^2$$ 4. **Step (c): A fifth pirate arrives at the center $P(0,0)$ and sets up a base. Calculate the percentage of land each original pirate must give up.** - The fifth pirate's base at the center creates new boundaries. - The original four pirates' cells are now divided by the captain's base. - The captain's cell is a smaller circle or polygon around the center. - Since the captain is at the center, the original four cells lose equal parts of their land. - The captain's cell area is a circle with radius equal to half the distance between the center and each pirate base (which is 8 km), so radius is 4 km. - Area of captain's cell: $$\pi \times 4^2 = 16\pi \approx 50.3 \text{ km}^2$$ - Total island area is $64\pi$, so captain's share is $\frac{16\pi}{64\pi} = \frac{1}{4} = 25\%$. - Each original pirate loses $\frac{25\%}{4} = 6.25\%$ of their land to the captain. **Final answers:** - (a) Pirates' bases coordinates: $(8,0)$, $(0,8)$, $(-8,0)$, $(0,-8)$ - (b) Area of each pirate's cell: $16\pi \approx 50.3$ km$^2$ - (c) Each original pirate gives up $6.25\%$ of their land to the captain.