ENGI 2422 Engineering Mathematics 2

Faculty of Engineering and Applied Science
2008 Winter

Problem Set 5   Questions

1. The position   x   of a particle moving along a wire is some function   x(t)   of the elapsed time   t.   The velocity   v   of the particle is given as a function of both x and t as

   v   =   e^–t cos t   –   x

   1. Use the chain rule to find the acceleration a(x, t) of the particle, (as a function of both x and t).
   2. Show that   x = e^–t sin t   satisfies the equation   v   =   e^–t cos t   –   x
   3. Hence express the acceleration as a function of time only.

2. The temperature   T   inside a material is modelled by

   T   =   10 r e^–r/10

   where   r   is the distance of the point (x, y, z) from the origin.

   1. Find the rate at which the temperature is [instantaneously] increasing at the point (4, 4, 7) when one is moving in the direction of   a = 2i + j – 2k .
   2. At what location(s) is the temperature not changing in any direction?
   3. At all other locations, in what direction is the temperature increasing most rapidly?

3. Determine the location and nature of all critical points of the function

   f (x, y)   =   xy (1 – xy²)

4. Determine the location and nature of all critical points of the function   z = f (x, y), where

   x² + 2 x + y² + 4 y + 2 z + 5   =   0

5. When a current   I   enters two resistors that are connected in parallel, with resistances   R₁   and   R₂, it splits into two currents   I₁   and   I₂   (with   I = I₁ + I₂) in such a way that the total power   P = R₁ I₁² + R₂ I₂²   is minimized.

   1. Express   I₁ and I₂   in terms of   R₁,   R₂   and   I.
   2. Hence determine a formula for the total resistance   R.

6. A window is to be constructed in the shape of a rectangle surmounted by an isosceles triangle.   In order to meet building code requirements for the room in which it is to be placed, the window must have a fixed glass area of 3 square metres.   Determine the dimensions of the window such that the cost of the materials is minimized.
   [Hint:   Since the glass area is fixed, the only way to accomplish the task is to minimize the amount of the material needed for the perimeter of the window.]

7. Determine the location and nature of all critical points for the function   h(x, y, z) = y   on the ellipse of intersection of the cone   f (x, y, z) = x² – y² + z² = 0   and the plane   g(x, y, z) = x + 3y + 2z = 3 .
   [Hint:   This is a Lagrange Multiplier problem with two constraints.]

8. Find the point on the sphere, radius 1, centre the origin, the sum of whose Cartesian coordinates is the greatest.

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