Question

A right circular cone has a base radius of $r$ and a height of $h.$ If the radius decrease by $20$ percent and the height increase by $10$ percent, which of the following is the resulting percent change in the volume of the cone?

• A. $10$% decrease
• B. $18.4$% decrease
• C. $12$% decrease
• D. $29.6$% decrease

Answer 1

Answer: (D)

radius $=r$ height $=h$

volume $=\frac{1}{3}\pi r^{2}h=V$

New radius $=r-\frac{20}{100}r=\frac{4}{5}r$ ($20\%$ decrease)

New height $=h+\frac{10}{100}h=\frac{11}{10}h$ ($10\%$ decrease)

New volume $=\frac{1}{3}\pi \left(\frac{4}{5}{r}^2\right)\left(\frac{11}{10}h\right)$

$⟹\frac{88}{125}\left(\frac{1}{3}\pi r^{2}h\right)=\frac{88}{125}V$

$=V-\frac{37}{125}V$

$=V-\left(29.6\%\right)$ of $V$

$\therefore 29.6\%$ decrease in volume.