DIRECTIONS: Please write complete, concise responses to the following problems. Show all relevant work on any calculations, and be sure to explain what you are doing. Please include this assignment sheet when you submit your work. 1. Differential Equations: A sphere with radius Im has temperature 15 C. It lies inside a concentric sphere with radius 2m and temperature 25 C. The temperature T(r) at a distance r from the common center of the spheres satisfies the following 2nd-order differential equation: d'T 2 dT dr2 + = 0 r dr If we let S = -, then S satisfies a Ist-order differential equation. Solve this equation by separation of variables; then, use the original substitution to find an expression for the temperature T(r) between the spheres. 2. Geometric Series: A certain ball has the property that each time it falls from a given height, it rebounds to 80% of its previous height. If we assume that air resistance is negligible, then the total distance that the ball travels during all its bounces will involve sums of geometric series. Assuming that the ball is dropped from an initial height of 10 feet, find this total vertical distance traveled, keeping in mind that the ball doesn't just travel downwards; it travels upwards as well. We can also calculate total time. In general, the time T that it takes for a ball to reach the ground, for any bounce, from a height H satisfies the relationship T = 2H How long does it take for the ball in this problem g to stop bouncing? Remember to keep in mind the total time for both upward and downward movement (Use g = 32ft/sec?)