Question

suppose planet as diameter is approximately 7 times planet bs and both bodies are spheres. what is the ratio of their volumes? the ratio of planet bs volume to planet as volume is (type the ratio as a simplified fraction)

Explanation:

Step1: Find the relationship between radii

Let the radius of planet B be \(r\). Since the diameter of planet A is 7 times that of planet B, and diameter \(d = 2r\), the radius of planet A, \(R=\frac{7\times(2r)}{2}=7r\).

Step2: Recall the volume formula for a sphere

The volume formula for a sphere is \(V=\frac{4}{3}\pi r^{3}\). The volume of planet B, \(V_{B}=\frac{4}{3}\pi r^{3}\), and the volume of planet A, \(V_{A}=\frac{4}{3}\pi R^{3}\). Substitute \(R = 7r\) into the formula for \(V_{A}\), we get \(V_{A}=\frac{4}{3}\pi(7r)^{3}=\frac{4}{3}\pi\times343r^{3}\).

Step3: Calculate the ratio of volumes

The ratio of planet B's volume to planet A's volume is \(\frac{V_{B}}{V_{A}}=\frac{\frac{4}{3}\pi r^{3}}{\frac{4}{3}\pi\times343r^{3}}\). Cancel out \(\frac{4}{3}\pi r^{3}\) from the numerator and denominator.

Answer:

\(\frac{1}{343}\)