If the diameter of base of a right circular cylinder is decreased by 10%, then volume of cylinder remains the same. Find the percentage increase in height.

Options

A. 0.2
B. 0.2345
C. 0.15
D. 0.2054

Answer

To solve the problem, we need to find the percentage increase in height of a right circular cylinder when its diameter is decreased by 10% while keeping the volume constant.

### Step-by-Step Solution:

1. **Define the Initial Diameter and Radius:**
- Let the initial diameter of the cylinder be \( D_1 = 100 \) cm.
- The initial radius \( R_1 \) is half of the diameter:
\[ R_1 = \frac{D_1}{2} = \frac{100}{2} = 50 \text{ cm} \]

2. **Calculate the Decreased Diameter and New Radius:**
- The diameter is decreased by 10%, so the new diameter \( D_2 \) is:
\[ D_2 = D_1 - 0.1 \times D_1 = 100 - 10 = 90 \text{ cm} \]
- The new radius \( R_2 \) is:
\[ R_2 = \frac{D_2}{2} = \frac{90}{2} = 45 \text{ cm} \]

3. **Set Up the Volume Equation:**
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
- For the initial cylinder:
\[ V_1 = \pi R_1^2 H_1 = \pi (50)^2 H_1 = 2500\pi H_1 \]
- For the new cylinder:
\[ V_2 = \pi R_2^2 H_2 = \pi (45)^2 H_2 = 2025\pi H_2 \]
- Since the volumes are equal (\( V_1 = V_2 \)):
\[ 2500\pi H_1 = 2025\pi H_2 \]

4. **Cancel \(\pi\) and Rearrange:**
- Cancel \(\pi\) from both sides:
\[ 2500 H_1 = 2025 H_2 \]
- Rearranging gives:
\[ H_2 = \frac{2500 H_1}{2025} \]

5. **Substitute Initial Height:**
- Let’s assume the initial height \( H_1 = 100 \) cm:
\[ H_2 = \frac{2500 \times 100}{2025} = \frac{250000}{2025} \approx 123.46 \text{ cm} \]

6. **Calculate the Change in Height:**
- The change in height \( \Delta H \) is:
\[ \Delta H = H_2 - H_1 = 123.46 - 100 = 23.46 \text{ cm} \]

7. **Calculate the Percentage Increase in Height:**
- The percentage increase in height is given by:
\[ \text{Percentage Increase} = \left( \frac{\Delta H}{H_1} \right) \times 100 = \left( \frac{23.46}{100} \right) \times 100 = 23.46\% \]

### Final Answer:
The percentage increase in height is approximately **23.46%**.