Question

if the radius of a cylinder is halved and the height remains the same, the volume will decrease by a factor of ______. a. 1/2 b. 1/8 c. 1/3 d. 1/4
the height of a cylinder with a volume of 314 cm³ and a radius of 5 cm is ______. a. 5 cm b. 10 cm c. 4 cm d. 15 cm

Explanation:

Step1: Recall volume formula for cylinder

The volume formula of a cylinder is $V = \pi r^{2}h$.

Step2: Analyze first - part change in volume

Let the original radius be $r$ and height be $h$, so original volume $V_1=\pi r^{2}h$. New radius is $\frac{r}{2}$ and height is $h$, new volume $V_2=\pi(\frac{r}{2})^{2}h=\pi\frac{r^{2}}{4}h$. The ratio $\frac{V_2}{V_1}=\frac{\pi\frac{r^{2}}{4}h}{\pi r^{2}h}=\frac{1}{4}$. So the volume decreases by a factor of $\frac{1}{4}$.

Step3: Solve second - part for height

Given $V = 314\ cm^{3}$, $r = 5\ cm$, from $V=\pi r^{2}h$, we have $h=\frac{V}{\pi r^{2}}$. Substitute $V = 314$, $r = 5$ and $\pi\approx3.14$ into the formula: $h=\frac{314}{3.14\times5^{2}}=\frac{314}{3.14\times25}=\frac{314}{78.5}=4\ cm$.

Answer:

a. $\frac{1}{4}$
c. $4\ cm$