Question

the moon has a volume of about $\frac{4}{3}pi(12^{3})^{3}$ cubic kilometers. the earth has a volume of about $\frac{4}{3}pi(16cdot20^{2})^{3}$ cubic kilometers. the radius of the earth is $\frac{u}{27}$ times as long as the radius of the moon. what is the value of $u$?

Explanation:

Step1: Recall the volume formula for a sphere

The volume formula for a sphere is $V = \frac{4}{3}\pi r^{3}$, where $r$ is the radius. For the moon, $V_m=\frac{4}{3}\pi(12^{3})^{3}$, so the radius of the moon $r_m = 12^{3}$. For the earth, $V_e=\frac{4}{3}\pi(16\cdot20^{2})^{3}$, so the radius of the earth $r_e=16\cdot20^{2}$.

Step2: Set up the ratio of the radii

We know that $r_e=\frac{u}{27}r_m$. Substitute $r_m = 12^{3}$ and $r_e=16\cdot20^{2}$ into the equation: $16\cdot20^{2}=\frac{u}{27}\times12^{3}$.
First, calculate $12^{3}=12\times12\times12 = 1728$ and $20^{2}=400$, so $16\times400=\frac{u}{27}\times1728$.
$6400=\frac{u\times1728}{27}$.

Step3: Solve for $u$

Cross - multiply: $6400\times27 = u\times1728$.
$u=\frac{6400\times27}{1728}$.
$6400\times27 = 172800$.
$u=\frac{172800}{1728}=100$.

Answer:

100