The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Explanation

Let diameter of the Earth is `d_(2)`.
We have, diameter of Moon `(d_(1))=(1)/(4)` diameter of the Earth
`d_(1)=(1)/(4)d_(2)`
Hence, the radius of the Moon `= (d_(1))/(2) = (d_(2))/(8)`
The raidius of the Earth `= (d_(2))/(2)`
`therefore` Volume of the Earth `(V_(1))=(4)/(3)pi ((d_(2))/(2))^(3)=(4)/(3)pi xx (d_(2)^(3))/(8)`
`therefore` Volume of the Moon `(V_(2))=(4)/(3)pi ((d_(2))/(8))^(3)=(4)/(3)pi(d_(2)^(3))/(512)`
`therefore` Required fraction `= (V_(1))/(V_(2))=((4)/(3)pi((d_(2)^(3))/(8)))/((4)/(3)pi((d_(2)^(3))/(512)))`
`rArr (V_(1))/(V_(2))=(512)/(8)`
`rArr (V_(1))/(V_(2))=(64)/(1)`