If the radius of a circle increases from 5 cm to 5.1 cm, find the increase in area.

Answer

To find the increase in the area of a circle when the radius increases from 5 cm to 5.1 cm, we can follow these steps:

### Step 1: Identify the initial and final radius
Let the initial radius \( r_1 = 5 \) cm and the final radius \( r_2 = 5.1 \) cm.

### Step 2: Write the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

### Step 3: Calculate the initial area
Using the initial radius \( r_1 \):
\[ A_1 = \pi (r_1)^2 = \pi (5)^2 = 25\pi \text{ cm}^2 \]

### Step 4: Calculate the final area
Using the final radius \( r_2 \):
\[ A_2 = \pi (r_2)^2 = \pi (5.1)^2 = \pi (26.01) \text{ cm}^2 \]

### Step 5: Calculate the increase in area
The increase in area \( \Delta A \) is given by:
\[ \Delta A = A_2 - A_1 = \pi (26.01) - 25\pi = \pi (26.01 - 25) = \pi (1.01) \text{ cm}^2 \]

### Step 6: Calculate the numerical value of the increase in area
Using the approximate value of \( \pi \approx 3.14 \):
\[ \Delta A \approx 3.14 \times 1.01 \approx 3.17 \text{ cm}^2 \]

### Final Answer
The increase in area when the radius of the circle increases from 5 cm to 5.1 cm is approximately \( 3.17 \text{ cm}^2 \).

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