How can I find the distance?

Question

Mountain pull. A large mountain can slightly affect the direction of “down” as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R = 2.00 km and density (mass per unit volume) 2.6 × 10^3 kg/m3. Assume also that we hang a 0.5 m plumb line at a distance of 3R from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?

I came to the conclusion that it needs to be G*M/ r^2*g=tan (theta)
I cannot find a way to solve for distance.

Answer 1

force gravity= GMearth*m/reart^2
force mountain= GMmountain*m/(2e3)^2

arctanTheta= forcemountain/forceEarth
= check this
= MassMountain/massEarth*(radiusearth/2000)^2

now you have theta. notice the angle is small, use the small angle approximation (sinT=tanT)
distance/.5=sinTheta
distance= .3*tanTheta, tan theta above.