Question

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:

• A. \( \pi^2 : 4 \)
• B. \( \pi : 4 \)
• C. \( \pi : 2 \)
• D. \( 1 : 4 \)
• E. \( 4 : \pi \)

Hint:

Remember, the ratio of areas for two objects with the same perimeter is influenced by their geometric shapes. For square and circle, the ratio simplifies in terms of \(\pi\).

Answer 1

The Correct Option is A

To find the ratio of the areas of Adu’s piece of land to Amu’s piece of land, we need to use the given information about their shapes and perimeters: 

1. Adu’s piece of land is a square, and Amu’s piece of land is a circle. Both have the same perimeter.
2. Let the side of Adu's square be \(s\). The perimeter of the square is given by \(4s\).
3. Let the radius of Amu's circle be \(r\). The perimeter (circumference) of the circle is given by \(2\pi r\).
4. Since the perimeters are equal, we have the equation: 
   \(4s = 2\pi r\) 
   Simplifying for \(s\), we get: 
   \(s = \frac{\pi r}{2}\)
5. The area of the square is: 
   \(A_{\text{square}} = s^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4}\)
6. The area of the circle is: 
   \(A_{\text{circle}} = \pi r^2\)
7. Now, the ratio of the area of Adu’s piece of land to Amu’s piece of land is: 
   \(\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{\frac{\pi^2 r^2}{4}}{\pi r^2} = \frac{\pi^2}{4\pi} = \frac{\pi}{4}\)

Therefore, the correct ratio of the areas is \(\pi^2 : 4\).

Answer 2

Step 1: Recall the formulae for the area of square and circle.
Let the perimeter of Adu's square land be \( P \). The perimeter of a square is given by \( P = 4 \times \text{side length} \), so the side length of the square is \( \frac{P}{4} \). The area of the square is \( \text{Area of square} = \left( \frac{P}{4} \right)^2 = \frac{P^2}{16} \). For Amu's circular land, the perimeter is the circumference, which is given by \( P = 2 \pi r \), where \( r \) is the radius. Thus, \( r = \frac{P}{2\pi} \). The area of the circle is \( \text{Area of circle} = \pi r^2 = \pi \left( \frac{P}{2\pi} \right)^2 = \frac{P^2}{4\pi} \).
Step 2: Calculate the ratio of areas.
The ratio of the areas of Adu’s square to Amu’s circle is: \[ \text{Ratio} = \frac{\frac{P^2}{16}}{\frac{P^2}{4\pi}} = \frac{1}{16} \times \frac{4\pi}{1} = \frac{\pi}{4} \]
Final Answer: \[ \boxed{\pi^2 : 4} \]