If the radius of a circle is increased by 19%, then by what percentage will its area increase?

Options

A. 27.89
B. 32.85
C. 11.91
D. 41.61

Answer

To solve the problem of how much the area of a circle increases when the radius is increased by 19%, we can follow these steps:

### Step 1: Understand the relationship between radius and area
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Calculate the initial area
Assuming the initial radius \( r \) is 10 cm, we can calculate the initial area:
\[ A_{\text{initial}} = \pi (10^2) = \pi \times 100 = 100\pi \text{ cm}^2 \]

### Step 3: Calculate the new radius after a 19% increase
To find the new radius after a 19% increase, we calculate:
\[ \text{New radius} = r + 0.19r = 1.19r \]
Substituting \( r = 10 \):
\[ \text{New radius} = 1.19 \times 10 = 11.9 \text{ cm} \]

### Step 4: Calculate the new area using the new radius
Now, we can calculate the new area with the new radius:
\[ A_{\text{new}} = \pi (11.9^2) = \pi \times 141.61 \approx 141.61\pi \text{ cm}^2 \]

### Step 5: Calculate the increase in area
The increase in area can be calculated as:
\[ \text{Increase in area} = A_{\text{new}} - A_{\text{initial}} = 141.61\pi - 100\pi = 41.61\pi \text{ cm}^2 \]

### Step 6: Calculate the percentage increase in area
To find the percentage increase in area, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase in area}}{A_{\text{initial}}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{41.61\pi}{100\pi} \right) \times 100 = \frac{41.61}{100} \times 100 = 41.61\% \]

### Final Answer
The area of the circle increases by **41.61%** when the radius is increased by 19%.

---