Question

the volume of a cylinder is 200π cm³ and its height is 8 cm. what is the radius of the base? a. 5 cm b. 4 cm c. 3 cm d. 10 cm if the radius of a cylinder is halved and the height remains the same, the volume will decrease by a factor of _______. a. 1/3 b. 1/8 c. 1/2 d. 1/4

Explanation:

Step1: Recall volume formula for cylinder

The volume formula of a cylinder is $V=\pi r^{2}h$, where $V$ is volume, $r$ is radius of the base and $h$ is height.

Step2: Solve for radius in first - question

Given $V = 200\pi\ cm^{3}$ and $h = 8\ cm$. Substitute into the formula: $200\pi=\pi r^{2}\times8$. Divide both sides by $8\pi$: $\frac{200\pi}{8\pi}=r^{2}$, so $r^{2}=25$, then $r = 5\ cm$.

Step3: Analyze volume change in second - question

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

Answer:

1. a. 5 cm
2. d. $\frac{1}{4}$