If the height of a right circular cone is increased by $200\%$ and the radius of the base is reduced by $50\%$, then the volume of the cone

### Concept & Proportionality The volume of a right circular cone is given by the formula: $$V = \frac{1}{3} \pi r^2 h$$ This means the volume is directly proportional to the height and the square of the radius ($V \propto r^2h$). Changes in volume can be found by applying the percentage changes as multiplying factors. ### Step-by-Step Solution * **Given:** * Initial Volume $V = \frac{1}{3} \pi r^2 h$ * New height $h' = h + (200\% \text{ of } h) = h + 2h = 3h$ * New radius $r' = r - (50\% \text{ of } r) = r - 0.5r = 0.5r = \frac{r}{2}$ * **Calculation:** 1. Substitute the new radius and height into the volume formula to find the new volume $V'$: $$V' = \frac{1}{3} \pi (r')^2 h'$$ $$V' = \frac{1}{3} \pi \left(\frac{r}{2}\right)^2 (3h)$$ 2. Expand and simplify the expression: $$V' = \frac{1}{3} \pi \left(\frac{r^2}{4}\right) (3h)$$ $$V' = \frac{3}{4} \left(\frac{1}{3} \pi r^2 h\right)$$ $$V' = \frac{3}{4} V$$ 3. Convert the fraction back to a percentage to find the change: $\frac{3}{4} = 75\%$ of the original volume. 4. Since the new volume is $75\%$ of the original, the volume has decreased by $100\% - 75\% = 25\%$. ### Exam Strategy & Shortcut Use scaling factors directly. Height increases by $200\% \rightarrow$ Factor is $3$. Radius decreases by $50\% \rightarrow$ Factor is $\frac{1}{2}$. Volume multiplier $= (\text{Radius Factor})^2 \times (\text{Height Factor}) = \left(\frac{1}{2}\right)^2 \times 3 = \frac{1}{4} \times 3 = \frac{3}{4}$. A multiplier of $\frac{3}{4}$ means a $25\%$ reduction. ### Common Pitfall A frequent error is forgetting to square the fractional change of the radius since the radius is squared in the volume formula. ### Final Answer Therefore, the correct answer is **decrease by $25\%$**.