The radius of a circle has been reduced from 9 cms to 7 cm. The appropriate percentage decrease in area is ----

Options

A. `31.5 %`
B. `39.5 %`
C. `34.5%`
D. `65.5%`

Answer

To find the percentage decrease in the area of a circle when the radius is reduced from 9 cm to 7 cm, we can follow these steps:

### Step 1: Calculate the area of the circle with the original radius (9 cm).
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
For the original radius \( r_1 = 9 \) cm:
\[ A_1 = \pi (9)^2 = \pi \times 81 = 81\pi \text{ cm}^2 \]

### Step 2: Calculate the area of the circle with the new radius (7 cm).
For the new radius \( r_2 = 7 \) cm:
\[ A_2 = \pi (7)^2 = \pi \times 49 = 49\pi \text{ cm}^2 \]

### Step 3: Calculate the decrease in area.
The decrease in area \( \Delta A \) is given by:
\[ \Delta A = A_1 - A_2 = 81\pi - 49\pi = (81 - 49)\pi = 32\pi \text{ cm}^2 \]

### Step 4: Calculate the percentage decrease in area.
The percentage decrease is calculated using the formula:
\[ \text{Percentage Decrease} = \left( \frac{\Delta A}{A_1} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Decrease} = \left( \frac{32\pi}{81\pi} \right) \times 100 \]
The \( \pi \) cancels out:
\[ \text{Percentage Decrease} = \left( \frac{32}{81} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Decrease} \approx 39.5\% \]

### Final Answer:
The appropriate percentage decrease in area is **39.5%**.

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