Question

If the radius of a circular region were decreased by 20 percent, the area of the circular region would decrease by what percent?

• A. 16%
• B. 20%
• C. 36%
• D. 40%
• E. 44%

Hint:

A quick way to solve percentage change problems is to use a scale factor. A 20% decrease means the new value is \(1 - 0.20 = 0.8\) times the old value. Since Area \(\propto\) radius\(^2\), the new area will be \((0.8)^2 = 0.64\) times the old area. A new value of 0.64 means a decrease of \(1 - 0.64 = 0.36\), or 36%.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires calculating the percentage change in the area of a circle that results from a percentage change in its radius. Since the area depends on the square of the radius, the percentage change in area will not be the same as the percentage change in the radius.
Step 2: Key Formula or Approach:
The area of a circle is given by the formula \(A = \pi r^2\), where \(r\) is the radius.
The formula for percentage decrease is: \(\frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100%\).
Step 3: Detailed Explanation:
Let the original radius be \(r_1\).
The original area is \(A_1 = \pi r_1^2\).
The radius is decreased by 20%. The new radius, \(r_2\), is \(100% - 20% = 80%\) of the original radius.
\[ r_2 = 0.80 \times r_1 \] Now, we calculate the new area, \(A_2\), using the new radius:
\[ A_2 = \pi r_2^2 = \pi (0.8 r_1)^2 = \pi (0.64 r_1^2) \] Since \(A_1 = \pi r_1^2\), we can write the new area in terms of the old area:
\[ A_2 = 0.64 A_1 \] This means the new area is 64% of the original area.
The decrease in area is the difference between the original area and the new area:
\[ \text{Decrease} = A_1 - A_2 = A_1 - 0.64 A_1 = 0.36 A_1 \] To express this decrease as a percentage, we multiply by 100.
\[ \text{Percentage Decrease} = 0.36 \times 100% = 36% \] Step 4: Final Answer:
The area of the circular region would decrease by 36%.