Question

If each shaded circular region in the figure above has radius 5, then the total area of the shaded regions is what fraction of the area of the square region? 

 

• A. \(\frac{\pi}{12}\)
• B. \(\frac{\pi}{36}\)
• C. \(\frac{\pi}{60}\)
• D. \(\frac{1}{6}\)
• E. \(\frac{1}{3}\)

Hint:

When simplifying a large fraction like 75/900, you don't need to find the greatest common divisor in one step. You can simplify incrementally. For example: 75/900 → divide by 5 → 15/180 → divide by 5 → 3/36 → divide by 3 → 1/12.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires us to find the ratio between the total area of three identical circles and the area of a square that contains them.
Step 2: Key Formula or Approach:
1. The formula for the area of a circle is \(A_{circle} = \pi r^2\).
2. The formula for the area of a square is \(A_{square} = s^2\).
3. We will calculate both areas using the given dimensions and then form the required fraction.
Step 3: Detailed Explanation:
1. Calculate the total area of the shaded circles:
- The radius \(r\) of each circle is given as 5.
- The area of a single circle is \(\pi \times 5^2 = 25\pi\).
- Since there are three identical circles, their total area is \(3 \times 25\pi = 75\pi\).
2. Calculate the area of the square:
- The side length \(s\) of the square is given as 30.
- The area of the square is \(s^2 = 30^2 = 900\).
3. Calculate the fraction:
- The fraction is the ratio of the total shaded area to the area of the square:
\[ \text{Fraction} = \frac{\text{Total Area of Circles}}{\text{Area of Square}} = \frac{75\pi}{900} \]
- Simplify the fraction by dividing numerator and denominator by 75:
\[ \frac{75\pi}{900} = \frac{1}{12} \pi = \frac{\pi}{12} \]
Step 4: Final Answer:
The total area of the shaded regions is \(\frac{\pi}{12}\) of the area of the square region.