If the radius of the Earth were increased by a factor 5, by what factor would its density have to be changed to keep the value of acceleration due to gravity the same?

Explanation

Let R, `rho` be the radius and density of Earth. Then the mass of the Earth.
`M=(4)/(3)piR^(3)rho`
`g=(GM)/(R^2)=(G)/(R^2)xx(4)/(3)piR^(3)rho`
`=(4)/(3)piGRrho`
`thereforerho=(3g)/(4piGR)`
When R' = 5R, then
`rho'=(3g)/(4piG(5R))=(1)/(5)(3g)/(4piGrho)=(rho)/(5)`
Thus the density of Earth would change by a factor `(1)/(5)`.