# Large Sphere & Sunny Day

It always helps to draw a figure of the problem. Starting from a point 10m from the bottom of the sphere draw a line at angle $\theta$ (where $\theta$ is the same as the angle of shadow from that 1 meter vertical stick) such that the line just touches the edge of the sphere. Now draw the radius from the center of the sphere to where the line touches the sphere - this radius line will be at angle $\theta$ from the vertical. The horizontal distance from the touch point to where the shadow ends 10 meters out is twice the vertical height of the touch point: $10-R \sin \theta = 2(R + R \cos \theta )$. You have values for $\sin \theta$ and $\cos \theta$ from the fact that $\theta = arctan(1/2)$, so you can solve for R.