Find the total surface area (in`cm^(2)`) of a right circular cylinder of diameter 42 cm. and height 14 cm.

Options

A. 4488
B. 4250
C. 4010
D. 4620

Answer

To find the total surface area (TSA) of a right circular cylinder, we can follow these steps:

### Step 1: Identify the given values
- Diameter of the cylinder = 42 cm
- Height of the cylinder = 14 cm

### Step 2: Calculate the radius
The radius (r) is half of the diameter.
\[ r = \frac{\text{Diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm} \]

### Step 3: Use the formula for total surface area of a cylinder
The formula for the total surface area (TSA) of a cylinder is:
\[ \text{TSA} = 2\pi rh + 2\pi r^2 \]
Where:
- \( r \) is the radius
- \( h \) is the height
- \( \pi \) can be approximated as \( \frac{22}{7} \)

### Step 4: Substitute the values into the formula
Substituting the values of \( r \) and \( h \) into the TSA formula:
\[ \text{TSA} = 2 \times \frac{22}{7} \times 21 \times 14 + 2 \times \frac{22}{7} \times (21)^2 \]

### Step 5: Calculate the curved surface area
First, calculate the curved surface area:
\[ \text{Curved Surface Area} = 2\pi rh = 2 \times \frac{22}{7} \times 21 \times 14 \]
Calculating this:
\[ = 2 \times \frac{22}{7} \times 21 \times 14 = 2 \times \frac{22 \times 21 \times 14}{7} \]
Calculating \( 22 \times 21 = 462 \), and then:
\[ 462 \times 14 = 6468 \]
Now divide by 7:
\[ \frac{6468}{7} = 924 \]
So, the curved surface area is:
\[ \text{Curved Surface Area} = 2 \times 924 = 1848 \text{ cm}^2 \]

### Step 6: Calculate the area of the circular bases
Now calculate the area of the two circular bases:
\[ \text{Area of Bases} = 2\pi r^2 = 2 \times \frac{22}{7} \times (21)^2 \]
Calculating \( 21^2 = 441 \):
\[ = 2 \times \frac{22}{7} \times 441 \]
Calculating this:
\[ = \frac{2 \times 22 \times 441}{7} = \frac{19452}{7} = 2778 \text{ cm}^2 \]

### Step 7: Add both areas to find the total surface area
Now, add the curved surface area and the area of the bases:
\[ \text{TSA} = \text{Curved Surface Area} + \text{Area of Bases} = 1848 + 2778 = 4626 \text{ cm}^2 \]

### Final Answer
The total surface area of the cylinder is:
\[ \text{TSA} = 4626 \text{ cm}^2 \]