A right circular cylinder has height as 18 cm and radius as 7 cm. The cylinder is cut in three equal parts (by 2 cuts parallel to base). What is the percentage increase in total surface area?

Options

A. 62
B. 56
C. 48
D. 52

Answer

To solve the problem step by step, we will calculate the total surface area of the cylinder before and after it is cut into three equal parts. Then, we will find the percentage increase in the total surface area.

### Step 1: Calculate the Total Surface Area of the Cylinder Before Cutting

The formula for the total surface area (TSA) of a right circular cylinder is given by:

\[ \text{TSA} = 2\pi rh + 2\pi r^2 \]

Where:
- \( r \) is the radius,
- \( h \) is the height.

Given:
- Height \( h = 18 \) cm,
- Radius \( r = 7 \) cm.

Substituting the values into the formula:

\[ \text{TSA} = 2\pi (7)(18) + 2\pi (7^2) \]

Calculating each part:

1. Calculate \( 2\pi rh \):
\[ 2\pi (7)(18) = 252\pi \]

2. Calculate \( 2\pi r^2 \):
\[ 2\pi (7^2) = 98\pi \]

Now, add both parts together:

\[ \text{TSA} = 252\pi + 98\pi = 350\pi \]

Using \( \pi \approx \frac{22}{7} \):

\[ \text{TSA} = 350 \times \frac{22}{7} = 1100 \text{ cm}^2 \]

### Step 2: Calculate the Total Surface Area After Cutting

When the cylinder is cut into three equal parts, we make two cuts. Each cut introduces a new circular surface area.

- Each circular base has an area of \( \pi r^2 \).
- After two cuts, there will be 4 circular surfaces (2 bases from the original cylinder and 2 new bases from the cuts).

The area contributed by the bases after cutting is:

\[ \text{Area from bases} = 4\pi r^2 \]

Substituting \( r = 7 \):

\[ \text{Area from bases} = 4\pi (7^2) = 4\pi (49) = 196\pi \]

Now, the total surface area after cutting is:

\[ \text{New TSA} = \text{Original TSA} + \text{Area from bases} \]
\[ \text{New TSA} = 350\pi + 196\pi = 546\pi \]

Using \( \pi \approx \frac{22}{7} \):

\[ \text{New TSA} = 546 \times \frac{22}{7} = 1716 \text{ cm}^2 \]

### Step 3: Calculate the Percentage Increase in Total Surface Area

The percentage increase in total surface area is calculated as follows:

\[ \text{Percentage Increase} = \left( \frac{\text{New TSA} - \text{Original TSA}}{\text{Original TSA}} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage Increase} = \left( \frac{1716 - 1100}{1100} \right) \times 100 \]
\[ = \left( \frac{616}{1100} \right) \times 100 \]
\[ = 56\% \]

### Final Answer:
The percentage increase in total surface area after cutting the cylinder into three equal parts is **56%**.