Question

sphere a has a diameter of 2 and is dilated by a scale factor of 3 to create sphere b. what is the ratio of the volume of sphere a to sphere b? 1:27 1:3 4:36 2:6

Explanation:

Step1: Find the radii of the spheres

The radius of sphere A, $r_A=\frac{d_A}{2}=\frac{2}{2} = 1$. Since sphere B is created by dilating sphere A by a scale - factor of 3, the radius of sphere B, $r_B = 3\times r_A=3\times1 = 3$.

Step2: Recall the volume formula for a sphere

The volume formula for a sphere is $V=\frac{4}{3}\pi r^{3}$. So the volume of sphere A, $V_A=\frac{4}{3}\pi r_A^{3}=\frac{4}{3}\pi(1)^{3}=\frac{4}{3}\pi$, and the volume of sphere B, $V_B=\frac{4}{3}\pi r_B^{3}=\frac{4}{3}\pi(3)^{3}=36\pi$.

Step3: Calculate the ratio of the volumes

The ratio of the volume of sphere A to sphere B is $\frac{V_A}{V_B}=\frac{\frac{4}{3}\pi}{36\pi}$. Canceling out $\pi$ and $\frac{4}{3}$ from the numerator and denominator, we get $\frac{1}{27}$.

Answer:

$1:27$