# Practice Exam - Math 13 - Feb 99'

1. Find the volume inside the sphere x2+y2+z2=4a2 and outside the cylinder x2+y2=2ax.

2. Using polar coordinates, evaluate

3. Use Lagrange multipliers to prove that the triangle with maximal area that has a given perimeter P is equilateral.[HINT: Use Heron's formula for the area : where and x,y and z are the lengths of the sides].

4. Let S be the surface

Find the point of S which is the closest to the xy plane.

5. Find the tangential and normal components of the acceleration of . Compute the curvature of the trajectory at time t=0.

Math 13 Winter 1999
1999-02-09