# Math Help - Large Sphere & Sunny Day

1. ## Large Sphere & Sunny Day

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow reaches out a distance of 10 m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of 2 m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)

2. ## Re: 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.