If diameter of a circle is decreased by `5%` by what percent its area will be decreased?

Options

A. `10.25%`
B. `9.75%`
C. `9%`
D. `10.36%`

Answer

To solve the problem of how much the area of a circle decreases when its diameter is decreased by 5%, we can follow these steps:

### Step-by-Step Solution

1. **Define the Original Diameter**:
Let the original diameter of the circle be \( D \).

2. **Calculate the New Diameter**:
If the diameter is decreased by 5%, the new diameter \( D' \) can be calculated as:
\[ D' = D - 0.05D = 0.95D \]

3. **Calculate the Original Radius**:
The original radius \( r \) is half of the original diameter:
\[ r = \frac{D}{2} \]

4. **Calculate the New Radius**:
The new radius \( r' \) is half of the new diameter:
\[ r' = \frac{D'}{2} = \frac{0.95D}{2} = 0.475D \]

5. **Calculate the Original Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the original radius:
\[ A = \pi \left(\frac{D}{2}\right)^2 = \pi \frac{D^2}{4} \]

6. **Calculate the New Area**:
The new area \( A' \) with the new radius is:
\[ A' = \pi (r')^2 = \pi \left(0.475D\right)^2 = \pi \left(\frac{0.95D}{2}\right)^2 = \pi \frac{(0.95D)^2}{4} = \pi \frac{0.9025D^2}{4} \]

7. **Calculate the Decrease in Area**:
The decrease in area \( \Delta A \) is given by:
\[ \Delta A = A - A' = \pi \frac{D^2}{4} - \pi \frac{0.9025D^2}{4} \]
Simplifying this:
\[ \Delta A = \pi \frac{D^2}{4} \left(1 - 0.9025\right) = \pi \frac{D^2}{4} \cdot 0.0975 \]

8. **Calculate the Percentage Decrease in Area**:
The percentage decrease in area is given by:
\[ \text{Percentage Decrease} = \left(\frac{\Delta A}{A}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \left(\frac{\pi \frac{D^2}{4} \cdot 0.0975}{\pi \frac{D^2}{4}}\right) \times 100 = 0.0975 \times 100 = 9.75\% \]

### Final Answer
The area of the circle will decrease by **9.75%** when the diameter is decreased by 5%.