Question

4. The moon’s diameter is approximately one-fourth the diameter of the earth. Compares the volumes of the moon and the earth.

Answer 1

Okay, here we have this:

Considering the provided information we obtain the following:

Let us take the diameter of the earth as d, so the diameter of the moon is d/4

Thus, the earth radius is d/2, and the moon radius is d/4/2=d/8.

Now, let's find the volume of the the earth and the moon separately to finally compare them:

`\begin{gathered} \text{Volume of earth=}(4)/(3)\pi r^3 \\ =(4)/(3)\pi((d)/(2))^3 \\ =(1)/(8)\cdot(4)/(3)\pi d^3 \\ =(1)/(6)\pi d^2 \end{gathered}`
`\begin{gathered} \text{Volume of moon=}(4)/(3)\pi r^3 \\ =(4)/(3)\pi((d)/(8))^3 \\ =(1)/(512)\cdot(4)/(3)\pi d^3 \\ =(1)/(384)\pi d^3 \end{gathered}`

Finally let's compare its volumes:

`\begin{gathered} \frac{Volume\text{ of moon}}{Volume\text{ of earth}}=((1)/(384)\pi d^3)/((1)/(6)\pi d^3) \\ =(1)/(64) \end{gathered}`

Finally we obtain that the volume of the moon is 1/64 of the volume of the earth.