Question

While a planet P rotates in a circle about its sun, a moon M rotates in a circle about the planet, and both motions are in a plane. Let's call the distance between M and P one Suppose the distance of P from the sun is 3.5×103 lunar units; the planet makes one revolution about the sun every 9 years, and the moon makes one rotation about the planet every 0.25 years. Choosing coordinates centered at the sun, so that, at time t=0 the planet is at (3.5×103, 0), and the moon is at (3.5×103, 1), then the location of the moon at time t, where t is measured in years, is (x(t),y(t)), where
x(t)=
y(t)=

Answer 1

Assuming 3.5x103 is:
\[3.5 x 10^3\]

To start, lets calculate the rotation of the planet around the sun. The period is
\[(2\pi/9)t\]
So for the x and y coordinate of the planet we get:
\[x(t) = 3.5*10^3 \cos((2\pi/9)t)\] ; \[y(t) = 3.5*10^3 \sin((2\pi/9)t)\]

For the moon, lets pretend the planet is stationary. The period for the moon around the planet is:
\[(2\pi/.25)t = 8\pi t\]

The x and y equations for the moon around the planet are:
\[x(t) = \cos(8\pi t) ; y(t) = \sin(8\pi t)\]

So, the complete equation for the motion of the moon around the sun, is the sum of these. We get:
\[x(t) = 3.5*10^3 \cos((2\pi/9)t) + cos(8\pi t)\]
\[y(t) = 3.5*10^3 \sin((2\pi/9)t) + sin(8\pi t)\]

Answer 2

says they r both incorrect !