If the ratio of total surface area to the curved surface area of a cylinder be 4:1, what is the ratio of radius to the height ?

Options

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

Answer

To solve the problem, we need to find the ratio of the radius (r) to the height (h) of a cylinder given that the ratio of the total surface area (TSA) to the curved surface area (CSA) is 4:1.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
- The curved surface area (CSA) of a cylinder is given by the formula:
\[ \text{CSA} = 2\pi rh \]

2. **Set Up the Ratio**:
- According to the problem, the ratio of TSA to CSA is given as:
\[ \frac{\text{TSA}}{\text{CSA}} = \frac{4}{1} \]
- Substituting the formulas for TSA and CSA, we have:
\[ \frac{2\pi r (r + h)}{2\pi rh} = \frac{4}{1} \]

3. **Simplify the Equation**:
- The \(2\pi\) terms in the numerator and denominator cancel out:
\[ \frac{r (r + h)}{rh} = 4 \]
- This simplifies to:
\[ \frac{r + h}{h} = 4 \]

4. **Cross-Multiply**:
- Cross-multiplying gives:
\[ r + h = 4h \]

5. **Rearrange the Equation**:
- Rearranging the equation, we get:
\[ r = 4h - h \]
- This simplifies to:
\[ r = 3h \]

6. **Find the Ratio**:
- To find the ratio of radius to height, we write:
\[ \frac{r}{h} = \frac{3h}{h} = 3 \]
- Therefore, the ratio of radius to height is:
\[ r:h = 3:1 \]

### Final Answer:
The ratio of the radius to the height of the cylinder is \(3:1\).