Question

The equation shows the relationship between a planet’s orbital period, T, and the planet’s mean distance from the sun, A, in astronomical units, AU. If planet Y is k times the mean distance from the sun as planet X, by what factor is the orbital period increased?

Answer 1

Answer choice D: K^3/2

Answer 2

If planet Y is k times the mean distance from the sun as planet X,

mean distance from the sun is A

`T^2= A^3`

k times the mean distance, we multiply A^3 with k^3

Multiply both sides by k^3

`k^3T^2= k^3A^3`

`k^3T^2= (kA)^3`

we need to get whole square on the left side to make the equivalent equation

`k^3 is k^{(3)/(2))^2`

So we replace k^3 by fractional exponent

`(k^{(3)/(2))^2T^2)= (kA)^3`

`(k^{(3)/(2))T)^2= (kA)^3`

Hence factor is
`k^{(3)/(2)}`