Question

Circle B's diameter was multiplied by 1.8. By what percent, approximately, was the area increased?

• A. 80%
• B. 125%
• C. 225%
• D. 325%
• E. 375%

Hint:

When a linear dimension (like radius, diameter, side length) of a 2D shape is scaled by a factor of \(k\), its area is scaled by \(k^2\). The percentage increase is \((k^2 - 1) \times 100%\). Here, \(k=1.8\), so the increase is \((1.8^2 - 1) \times 100% = (3.24 - 1) \times 100% = 224%\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The area of a two-dimensional shape scales with the square of its linear dimensions. This means if a linear dimension like the radius or diameter is multiplied by a factor \(k\), the area is multiplied by a factor of \(k^2\).

Step 2: Key Formula or Approach:
The area of a circle is given by \( A = \pi r^2 \), where \(r\) is the radius. Since diameter \(d = 2r\), we can also write the area in terms of diameter: \( A = \pi (d/2)^2 = \frac{\pi d^2}{4} \).
The formula for percentage increase is: \[ \text{Percentage Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100% \]

Step 3: Detailed Explanation:
Let the original diameter be \(d_{old}\) and the original area be \(A_{old}\). \[ A_{old} = \frac{\pi d_{old}^2}{4} \] The new diameter, \(d_{new}\), is 1.8 times the old diameter: \[ d_{new} = 1.8 \times d_{old} \] Now, let's find the new area, \(A_{new}\), using the new diameter: \[ A_{new} = \frac{\pi d_{new}^2}{4} = \frac{\pi (1.8 \times d_{old})^2}{4} = \frac{\pi (1.8^2 \times d_{old}^2)}{4} \] \[ A_{new} = 1.8^2 \times \left( \frac{\pi d_{old}^2}{4} \right) = 1.8^2 \times A_{old} \] Calculate the scaling factor for the area: \[ 1.8^2 = 1.8 \times 1.8 = 3.24 \] So, the new area is 3.24 times the old area: \[ A_{new} = 3.24 \times A_{old} \] Now, we calculate the percentage increase: \[ \text{Increase} = \frac{A_{new} - A_{old}}{A_{old}} \times 100% = \frac{3.24 A_{old} - A_{old}}{A_{old}} \times 100% \] \[ \text{Increase} = \frac{2.24 A_{old}}{A_{old}} \times 100% = 2.24 \times 100% = 224% \] The question asks for the approximate percentage, and 224% is approximately 225%.
Step 4: Final Answer
The area was increased by approximately 225%.