Math 13, Winter 1998

*Date:* Thursday, 11 February; 19:00 - 20:20

Instructions:

- Complete all 4 questions. You may answer the questions in any order.
- Explain all your answers. Hand in all your working notes.
- You have 80 minutes to complete the exam.

- 1.
- (a)
- (20 points) Let be a positive number. Find the maximum of subject to the constraint that .
- (b)
- (5 points) What is the maximum possible volume of a cylinder contained inside a sphere of radius ?

- 2.
- (a)
- (5 points) Draw the region and in the -plane.
- (b)
- (20 points) Use polar coordinates to find the volume of the solid region lying above and below the surface . (20 points)

- 3.
- An ant is walking on a thin metal plate that occupies the region
- (a)
- (10 points) Suppose the ant is at the origin. In which direction should the ant move, so that its temperature decreases as quickly as possible?
- (b)
- (15 points) Where on the plate is the temperature the highest? What is the maximum temperature of any point on the plate? Justify your answer.

- 4.
- A particle is travelling so that its position in -space at
time
is given by
- (a)
- (10 points) Find the velocity, acceleration, and speed of the particle at time .
- (b)
- (10 points) Find the tangential component of acceleration of the particle at time .
- (c)
- (5 points) Find the length of the acceleration vector at time
. Use this to show that the curvature of the particle's trajectory at
time
is