Question

sure! lets create a problem related to a sphere, which is another shape thats tested on the sat. problem: a sphere has a volume of $\frac{32pi}{3}$. if the radius of the sphere is halved, what is the new volume of the sphere? choices: a. $\frac{4pi}{3}$ b. $\frac{8pi}{3}$ c. $8pi$ d. $16pi$

Explanation:

Step1: Recall volume formula for sphere

The volume formula of a sphere is $V = \frac{4}{3}\pi r^{3}$. Given $V=\frac{32\pi}{3}$, we have $\frac{4}{3}\pi r^{3}=\frac{32\pi}{3}$.

Step2: Solve for original radius

Dividing both sides of $\frac{4}{3}\pi r^{3}=\frac{32\pi}{3}$ by $\frac{4\pi}{3}$, we get $r^{3}=8$, so $r = 2$.

Step3: Find new radius

The new radius $r_{new}=\frac{r}{2}=\frac{2}{2}=1$.

Step4: Calculate new volume

Using the volume formula with the new - radius, $V_{new}=\frac{4}{3}\pi r_{new}^{3}=\frac{4}{3}\pi(1)^{3}=\frac{4\pi}{3}$.

Answer:

A. $\frac{4\pi}{3}$