The ratio of the total surface area to the lateral surface area of a cylineder with base radius 80 cm and height 20 cm is

Options

A. `2 :1`
B. `3:1`
C. `4:1`
D. `5:1`

Explanation

To find the ratio of the total surface area to the lateral surface area of a cylinder with a base radius of 80 cm and a height of 20 cm, we will follow these steps:

### Step 1: Understand the formulas
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
where \( r \) is the radius and \( h \) is the height.

The lateral surface area (LSA) of a cylinder is given by the formula:
\[ \text{LSA} = 2\pi rh \]

### Step 2: Substitute the values into the formulas
Given:
- Radius \( r = 80 \) cm
- Height \( h = 20 \) cm

**Calculate the Total Surface Area:**
\[ \text{TSA} = 2\pi r (r + h) = 2\pi (80) (80 + 20) = 2\pi (80)(100) \]

**Calculate the Lateral Surface Area:**
\[ \text{LSA} = 2\pi rh = 2\pi (80)(20) \]

### Step 3: Calculate the TSA and LSA
**Total Surface Area:**
\[ \text{TSA} = 2\pi (80)(100) = 16000\pi \, \text{cm}^2 \]

**Lateral Surface Area:**
\[ \text{LSA} = 2\pi (80)(20) = 3200\pi \, \text{cm}^2 \]

### Step 4: Find the ratio of TSA to LSA
Now, we will find the ratio of the total surface area to the lateral surface area:
\[ \text{Ratio} = \frac{\text{TSA}}{\text{LSA}} = \frac{16000\pi}{3200\pi} \]

### Step 5: Simplify the ratio
The \(\pi\) cancels out:
\[ \text{Ratio} = \frac{16000}{3200} = 5 \]

Thus, the ratio of the total surface area to the lateral surface area is:
\[ 5 : 1 \]

### Final Answer
The ratio of the total surface area to the lateral surface area of the cylinder is \( 5 : 1 \).

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