Question

@jcpd910 >>> The equation T^2 = A^3 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

I'm sorry, I have a massive problem picturing things like this in my head, so I need an image. :/

Answer 2

If \[T^{2}=A^{2}\]
lets assume that planet X has A astronomical units for a mean distance from the sun.

That means that planet Y has a mean distance of KA.

So\[T^{2}=(KA)^{2}=K^{2}A^{2}\]for planet Y.

Answer 3

Oops I thought it was T^2, the only thing that changes is\[T^{2}=(KA)^{3}=K^{3}A^{3}\]

Answer 4

T^2 = a^3

Answer 5

Yes I just typed it wrong...again.

Answer 6

so the ans is

Answer 7

What would the orbital period be for planet X in this situation?

Answer 8

hm 1/2

Answer 9

You would solve for T in the equation \[T^{2}=A^{3}\]

Answer 10

ooh 2/3

Answer 11

If \[T^{2}=A^{3}\]then to solve for T, we would take the square root of both sides.

Answer 12

i still got 2/3

Answer 13

Show me your work. Your answer should be in terms of A.

Answer 14

i did...\[\sqrt{T^2} = \sqrt{a^3}\]

Answer 15

then i came up with 3/2

Answer 16

3/2 is not the answer, it is the exponent. Your square root operation was correct. But...\[\sqrt{T^{2}}=\sqrt{A^{3}}\]\[T=A^{3/2}\]

Answer 17

ooh i did do it correct but i forgot the A = T ooh ok thanks

Answer 18

And also \[A=T^{2/3}\]
Now you need to solve for A in the equation with the K in it.

Answer 19

I'm sorry, solve for T.

Answer 20

i was about to say where did that K come from

Answer 21

It does have a K in it, remember your problem? K is the multiple for the distance of planet Y.
The equation is\[T^{2}=K^{3}A^{3}\]

Answer 22

Solve for T, which is the orbital period

Answer 23

oh i for got

Answer 24

ooh i have no clue how to go about that with the ^3 thing

Answer 25

\[T^{2}=K^{3}A^{3}\]\[\sqrt{T^{2}}=\sqrt{K^{3}A^{3}}=\sqrt{K^{3}}\sqrt{A^{3}}\]\[T=K^{3/2}A^{3/2}\]

Answer 26

So for planet X, the orbital period, T, would be:\[T=A^{3/2}\]
And for planet Y the orbital period would be:\[T=K^{3/2}A^{3/2}\]

Answer 27

oh i see

Answer 28

So by what factor is planet Y's orbital period larger than that of planet X?

Answer 29

ok im back

Answer 30

Okay. All you need to do is interpret the equations to get your answer.

Answer 31

would u add both the tops & bottums like 3+3 & 2+2

Answer 32

The question is basically asking you, "What would you multiply the orbital period of X by to get the orbital period of Y?"

Answer 33

The orbital period of X is A^{3/2}
and the period of Y is K^{3/2}A^{3/2}