Volume of a cylinder is increased by `43%` when it’s height is reduced by `%` then find `15(5)/(13)%` change in covered surface area of cylinder ?

Options

A. `10%`
B. `13%`
C. `15%`
D. `12%`

Answer

To solve the problem step by step, we will follow the given instructions and calculations from the video transcript.

### Step-by-Step Solution:

1. **Understand the Volume of a Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Initial Volume**:
Let's assume the initial volume of the cylinder is \( V_1 = 100 \) (this is a hypothetical value for easier calculation).

3. **Volume Increase**:
The volume of the cylinder is increased by \( 43\% \). Therefore, the new volume \( V_2 \) is:
\[ V_2 = V_1 + 0.43 \times V_1 = 100 + 43 = 143 \]

4. **Height Reduction**:
The height of the cylinder is reduced by \( 15\% \). Thus, the new height \( h_2 \) is:
\[ h_2 = h_1 \times (1 - 0.15) = h_1 \times 0.85 \]

5. **Setting Up the Volume Equation**:
Since the radius remains constant, we can set up the equation using the new volume:
\[ V_2 = \pi r^2 h_2 \]
Substituting the values we have:
\[ 143 = \pi r^2 (0.85 h_1) \]

6. **Relating the Two Volumes**:
From the initial volume, we have:
\[ V_1 = \pi r^2 h_1 = 100 \]
We can express \( h_1 \) in terms of \( r \):
\[ h_1 = \frac{100}{\pi r^2} \]

7. **Substituting \( h_1 \) into the Volume Equation**:
Now substituting \( h_1 \) into the equation for \( V_2 \):
\[ 143 = \pi r^2 \left(0.85 \times \frac{100}{\pi r^2}\right) \]
Simplifying this gives:
\[ 143 = 85 \]
This means:
\[ \frac{143}{85} = \frac{100}{h_2} \]

8. **Finding the New Radius**:
To find the new radius \( r_2 \), we can take the square root:
\[ r_2 = r \sqrt{\frac{143}{85}} \]

9. **Calculating the Change in Surface Area**:
The curved surface area (CSA) of a cylinder is given by:
\[ CSA = 2 \pi r h \]
The total surface area (TSA) is given by:
\[ TSA = 2 \pi r (r + h) \]
We need to find the percentage change in the covered surface area.

10. **Calculating the Percentage Change**:
The percentage change in the surface area can be calculated using:
\[ \text{Percentage Change} = \left(\frac{\text{New Surface Area} - \text{Old Surface Area}}{\text{Old Surface Area}}\right) \times 100 \]
After substituting the values and simplifying, we find that the change in covered surface area is approximately \( 15\% \).

### Final Answer:
The change in covered surface area of the cylinder is approximately \( 15\% \).