The radius of a cylinder is 10 cm and height is 4 cm. The number of centimetres that may be added either to the radius or to the height to get the same increase in the volume of the cylinder is

Options

A. 5 cm
B. 4 cm
C. 25 cm
D. 16 cm

Answer

To solve the problem, we need to determine the value of \( x \) that can be added to either the radius or the height of a cylinder, such that the volume increase is the same in both cases.

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius \( r = 10 \) cm
- Height \( h = 4 \) cm

2. **Write the formula for the volume of a cylinder:**
\[ V = \pi r^2 h \]

3. **Calculate the initial volume of the cylinder:**
\[ V = \pi (10)^2 (4) = \pi (100)(4) = 400\pi \text{ cm}^3 \]

4. **Consider the first case where we increase the radius:**
- New radius = \( 10 + x \)
- New volume = \( V_1 = \pi (10 + x)^2 (4) \)

5. **Consider the second case where we increase the height:**
- New height = \( 4 + x \)
- New volume = \( V_2 = \pi (10)^2 (4 + x) = \pi (100)(4 + x) \)

6. **Set the two volumes equal to each other:**
\[ \pi (10 + x)^2 (4) = \pi (100)(4 + x) \]

7. **Cancel out \( \pi \) from both sides:**
\[ (10 + x)^2 (4) = 100(4 + x) \]

8. **Expand both sides:**
- Left side:
\[ 4(10 + x)^2 = 4(100 + 20x + x^2) = 400 + 80x + 4x^2 \]
- Right side:
\[ 400 + 100x \]

9. **Set the expanded equations equal:**
\[ 400 + 80x + 4x^2 = 400 + 100x \]

10. **Subtract \( 400 \) from both sides:**
\[ 80x + 4x^2 = 100x \]

11. **Rearrange the equation:**
\[ 4x^2 + 80x - 100x = 0 \implies 4x^2 - 20x = 0 \]

12. **Factor out \( 4x \):**
\[ 4x(x - 5) = 0 \]

13. **Set each factor to zero:**
- \( 4x = 0 \) gives \( x = 0 \) (not valid in this context)
- \( x - 5 = 0 \) gives \( x = 5 \)

14. **Conclusion:**
The value of \( x \) that can be added to either the radius or the height to achieve the same increase in volume is \( \boxed{5} \) cm.