Q3
Adu and Amu have bought two pieces of land on the Moon from an e-store. Both the pieces of land have the same perimeters, but Adu’s piece of land is in the shape of a square, while Amu’s piece of land is in the shape of a circle.
The ratio of the areas of Adu’s piece of land to Amu’s piece of land is:
Answer: B
Asked: the ratio of areas of a square and a circle that have the same perimeter.
Let the common perimeter be \(L\).
- Square: side \(= \frac{L}{4}\), area \(= \frac{L^2}{16}\)
- Circle: \(2\pi r = L \Rightarrow r = \frac{L}{2\pi}\), area \(= \pi r^2 = \frac{L^2}{4\pi}\)
- Ratio \(= \frac{L^2/16}{L^2/(4\pi)} = \frac{4\pi}{16} = \frac{\pi}{4}\)
So Adu's area : Amu's area \(= \pi : 4\). (As expected, the circle encloses more area for the same perimeter, since \(\pi < 4\).)
Answer: \(\pi : 4\)