The area of a circle is increased by 22 cm' when its radius is increased by 1 cm. The original radius of the circle is

Options

A. 6cm
B. 3.2cm
C. 3cm
D. 3.5cm

Answer

To find the original radius of the circle, we will follow these steps:

### Step 1: Understand the problem
We know that when the radius of a circle is increased by 1 cm, the area increases by 22 cm². We need to find the original radius of the circle.

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

### Step 3: Set up the equation for the original and new area
Let the original radius be \( r \) cm. The original area \( A_1 \) is:
\[ A_1 = \pi r^2 \]
When the radius is increased by 1 cm, the new radius becomes \( r + 1 \) cm. The new area \( A_2 \) is:
\[ A_2 = \pi (r + 1)^2 \]

### Step 4: Set up the equation based on the increase in area
According to the problem, the increase in area is 22 cm², so we can write:
\[ A_2 - A_1 = 22 \]
Substituting the expressions for \( A_1 \) and \( A_2 \):
\[ \pi (r + 1)^2 - \pi r^2 = 22 \]

### Step 5: Simplify the equation
Factor out \( \pi \):
\[ \pi \left( (r + 1)^2 - r^2 \right) = 22 \]
Now, simplify \( (r + 1)^2 - r^2 \):
\[ (r + 1)^2 = r^2 + 2r + 1 \]
So,
\[ (r + 1)^2 - r^2 = 2r + 1 \]
Thus, we have:
\[ \pi (2r + 1) = 22 \]

### Step 6: Solve for \( r \)
Now, divide both sides by \( \pi \):
\[ 2r + 1 = \frac{22}{\pi} \]
Using \( \pi \approx 3.14 \):
\[ 2r + 1 = \frac{22}{3.14} \approx 7.006 \]
Subtract 1 from both sides:
\[ 2r = 7.006 - 1 = 6.006 \]
Now, divide by 2:
\[ r = \frac{6.006}{2} \approx 3.003 \]

### Step 7: Round the answer
Since we are looking for the original radius in whole centimeters, we can approximate:
\[ r \approx 3 \text{ cm} \]

### Final Answer
The original radius of the circle is approximately **3 cm**.

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