The radius of a solid right circular cylinder increases by 20% and its height decreases by 20%. Find the percentage change in its volume.

Answer

To find the percentage change in the volume of a solid right circular cylinder when its radius increases by 20% and its height decreases by 20%, we can follow these steps:

### Step-by-Step Solution:

1. **Assume Initial Dimensions:**
Let the initial radius of the cylinder be \( r \) and the initial height be \( h \).

2. **Calculate Initial Volume:**
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]

3. **Determine New Radius:**
Since the radius increases by 20%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.20r = 1.20r = \frac{6r}{5} \]

4. **Determine New Height:**
Since the height decreases by 20%, the new height \( h' \) can be calculated as:
\[ h' = h - 0.20h = 0.80h = \frac{4h}{5} \]

5. **Calculate New Volume:**
The new volume \( V' \) with the new dimensions is:
\[ V' = \pi (r')^2 (h') = \pi \left(\frac{6r}{5}\right)^2 \left(\frac{4h}{5}\right) \]
Simplifying this:
\[ V' = \pi \left(\frac{36r^2}{25}\right) \left(\frac{4h}{5}\right) = \pi \frac{144r^2h}{125} \]

6. **Find Change in Volume:**
The change in volume \( \Delta V \) is:
\[ \Delta V = V' - V = \pi \frac{144r^2h}{125} - \pi r^2h \]
Factor out \( \pi r^2h \):
\[ \Delta V = \pi r^2h \left(\frac{144}{125} - 1\right) = \pi r^2h \left(\frac{144 - 125}{125}\right) = \pi r^2h \left(\frac{19}{125}\right) \]

7. **Calculate Percentage Change in Volume:**
The percentage change in volume is given by:
\[ \text{Percentage Change} = \frac{\Delta V}{V} \times 100 = \frac{\pi r^2h \left(\frac{19}{125}\right)}{\pi r^2h} \times 100 \]
The \( \pi r^2h \) terms cancel out:
\[ \text{Percentage Change} = \frac{19}{125} \times 100 = 15.2\% \]

### Final Answer:
The percentage change in the volume of the cylinder is **15.2%**.

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