If diameter of a circle is decreased by 5%, by what per cent 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. **Understand the Problem**:
We need to find the percentage decrease in the area of a circle when its diameter is decreased by 5%.

2. **Initial Diameter**:
Let's assume the initial diameter of the circle is \( D = 100 \) units (this is a convenient choice to simplify calculations).

3. **Calculate the Decreased Diameter**:
The diameter decreases by 5%, so we calculate the new diameter:
\[ \text{New Diameter} = D - 0.05D = 100 - 5 = 95 \text{ units} \]

4. **Calculate the Initial Radius**:
The radius \( r \) is half of the diameter:
\[ r = \frac{D}{2} = \frac{100}{2} = 50 \text{ units} \]

5. **Calculate the New Radius**:
The new radius after the decrease in diameter is:
\[ r' = \frac{\text{New Diameter}}{2} = \frac{95}{2} = 47.5 \text{ units} \]

6. **Calculate the Initial Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Therefore, the initial area is:
\[ A = \pi (50)^2 = 2500\pi \]

7. **Calculate the New Area**:
The new area after the decrease in diameter is:
\[ A' = \pi (47.5)^2 \]
Calculating \( (47.5)^2 \):
\[ (47.5)^2 = 2256.25 \]
Thus, the new area is:
\[ A' = 2256.25\pi \]

8. **Calculate the Decrease in Area**:
The decrease in area \( \Delta A \) is:
\[ \Delta A = A - A' = 2500\pi - 2256.25\pi = (2500 - 2256.25)\pi = 243.75\pi \]

9. **Calculate the Percentage Decrease in Area**:
The percentage decrease in area is given by:
\[ \text{Percentage Decrease} = \left(\frac{\Delta A}{A}\right) \times 100 = \left(\frac{243.75\pi}{2500\pi}\right) \times 100 \]
Simplifying this:
\[ = \left(\frac{243.75}{2500}\right) \times 100 = 9.75\% \]

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