If the curved surface area of a solid right circular cylinder of height h and radius r is one-third of its total surface area, then h = (1/k)r, find k.

Answer

To solve the problem, we need to establish the relationship between the curved surface area (CSA) and the total surface area (TSA) of a right circular cylinder.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for the Curved Surface Area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
- The formula for the Total Surface Area (TSA) of a cylinder is given by:
\[ \text{TSA} = 2\pi r(h + r) \]

2. **Set Up the Equation**:
- According to the problem, the CSA is one-third of the TSA:
\[ 2\pi rh = \frac{1}{3} \times 2\pi r(h + r) \]

3. **Simplify the Equation**:
- We can cancel \(2\pi r\) from both sides (assuming \(r \neq 0\)):
\[ h = \frac{1}{3}(h + r) \]

4. **Multiply Both Sides by 3**:
- To eliminate the fraction, multiply both sides by 3:
\[ 3h = h + r \]

5. **Rearrange the Equation**:
- Rearranging gives:
\[ 3h - h = r \]
\[ 2h = r \]

6. **Express h in Terms of r**:
- Now, we can express \(h\) in terms of \(r\):
\[ h = \frac{1}{2}r \]

7. **Relate h to k**:
- The problem states that \(h = \frac{1}{k}r\). From our previous result, we have:
\[ \frac{1}{2}r = \frac{1}{k}r \]

8. **Compare the Coefficients**:
- Since \(r\) is not zero, we can compare the coefficients:
\[ \frac{1}{2} = \frac{1}{k} \]

9. **Solve for k**:
- Taking the reciprocal of both sides gives:
\[ k = 2 \]

### Final Answer:
Thus, the value of \(k\) is \(2\).