Let's solve the problem step-by-step:
1. Initial Setup:
- Start with a circle that has an initial radius, [tex]\( r_1 \)[/tex]. For simplification, let's assume [tex]\( r_1 \)[/tex] to be 1 unit. The result will be general and applicable for any radius since we are looking for a percentage increase.
2. Increased Radius:
- The radius increases by 50%. Therefore, the new radius, [tex]\( r_2 \)[/tex], can be calculated as:
[tex]\[
r_2 = r_1 + 0.5 \times r_1 = 1 + 0.5 = 1.5 \text{ units}
\][/tex]
3. Area of the Initial Circle:
- The area of a circle is given by the formula [tex]\( \text{Area} = \pi r^2 \)[/tex].
- Substituting the initial radius:
[tex]\[
\text{Area}_1 = \pi (1)^2 = \pi
\][/tex]
4. Area of the Increased Circle:
- Using the new radius [tex]\( r_2 \)[/tex]:
[tex]\[
\text{Area}_2 = \pi (1.5)^2 = \pi \times 2.25 = 2.25\pi
\][/tex]
5. Calculate the Percentage Increase in Area:
- To find the percentage increase in the area, use the formula:
[tex]\[
\text{Percentage Increase} = \left( \frac{\text{New Area} - \text{Initial Area}}{\text{Initial Area}} \right) \times 100
\][/tex]
- Substituting the values:
[tex]\[
\text{Percentage Increase} = \left( \frac{2.25\pi - \pi}{\pi} \right) \times 100
\][/tex]
- Simplify the expression:
[tex]\[
\text{Percentage Increase} = \left( \frac{2.25\pi - \pi}{\pi} \right) \times 100 = \left( \frac{1.25\pi}{\pi} \right) \times 100 = 1.25 \times 100 = 125\%
\][/tex]
Therefore, the area of the circle increases by 125%.
