The radius of a cylinder increases by 20% and its height increases by 10%, by what percent will its volume increase?

Question: The radius of a cylinder increases by 20% and its height increases by 10%, by what percent will its volume increase ?

• 20%
• 30.8%
• 45.5%
• 58.4%

Answer: 

$$V=\pi r^{2}h$$ $$V_1 = \pi r^2 h$$

After the changes:

• The radius increases by 20%, so the new radius is $1.2r$.

• The height increases by 10%, so the new height is $1.1h$.

The new volume is:

$$V_2 = \pi (1.2r)^2 (1.1h)$$ $$V_2 = \pi (1.44r^2) (1.1h)$$ $$V_2 = \pi r^2 h \times 1.584$$ $$V_2 = V_1 \times 1.584$$

The percentage increase in volume is:

$$\left( \frac{V_2 - V_1}{V_1} \right) \times 100 = (1.584 - 1) \times 100$$ $$=58.4\mathrm{\%}$$

**The volume increases by 58.4%.

Additional Method :

$$\text{Total Increase} \approx a + b + \frac{a \times b}{100}$$

• $a = 20\%$ (increase in radius)

• $b = 10\%$ (increase in height)

Since the volume depends on the square of the radius, we take twice the percentage increase in radius:

$$a' = 2 \times 20 = 40\%$$

The new volume is:

$$V_2 = \pi (1.2r)^2 (1.1h)$$ $$V_2 = \pi (1.44r^2) (1.1h)$$ $$V_2 = \pi r^2 h \times 1.584$$ $$V_2 = V_1 \times 1.584$$

**Thus, the volume increases by 54% (approximate).
**The exact increase using the full calculation was 58.4%, but this shortcut gives a quick and close estimate.

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