If the radius of a cylinder is decreased by 50% and the height is increased by 50% to form a new cylinder, the volume will be decreased by

### Concept & Successive Percentage Change The volume of a cylinder is $V = \pi r^2 h$. Because the radius is squared in the formula, a percentage change in the radius applies twice (or quadratically), while a percentage change in height applies only once linearly. ### Step-by-Step Solution 1. **Define the original volume:** - Let original radius $= r$, original height $= h$ - $V_1 = \pi r^2 h$ 2. **Calculate the new dimensions:** - Radius decreased by $50\% \Rightarrow r_2 = 0.5r = \frac{1}{2} r$ - Height increased by $50\% \Rightarrow h_2 = 1.5h = \frac{3}{2} h$ 3. **Calculate the new volume:** - $V_2 = \pi (0.5r)^2 (1.5h)$ - $V_2 = \pi (0.25r^2) (1.5h) = 0.375 \pi r^2 h$ - Equivalently in fractions: $V_2 = \pi \left(\frac{1}{4} r^2\right) \left(\frac{3}{2} h\right) = \frac{3}{8} \pi r^2 h$ 4. **Determine the percentage decrease:** - Decrease in volume = $V_1 - V_2 = 1 - \frac{3}{8} = \frac{5}{8}$ of the original volume. - Percentage decrease = $\frac{5}{8} \times 100\% = 62.5\%$ ### Exam Strategy & Shortcut Use scaling factors: $(0.5)^2 \times (1.5) = 0.25 \times 1.5 = 0.375$. This means the new volume is $37.5\%$ of the original. The decrease is $100\% - 37.5\% = 62.5\%$. ### Common Pitfall Treating the percentage changes linearly and simply adding $-50\% - 50\% + 50\% = -50\%$. You must account for the radius term being squared. ### Final Answer Therefore, the correct answer is **62.5%**.