Math 103 Test 2

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Due October 27, 1999

Instructions

This stage of the exam is due in class, Wednesday, October 27, 1999. I will not take late exams in my mailbox unless you have specifically discussed it with me.
Do this stage of the exam on your own paper. Please remember to staple your work before you come to class.
Organize your work in an unambiguous order. Show all necessary steps.
You may use a calculator on this stage of the exam where appropriate, but be sure to show the work leading up to your calculations. Do not use your calculator for vector operations.
You may use your notes and your textbook on this stage of the exam.
Do not discuss any stage of the exam with any other person besides your instructor before all stages are complete and handed in.
Good luck!

1.

(8%) Match the following functions with their level curves. In some cases graphs of the functions have been provided, but in other cases they have not.

(a): z=-(x²+y^2/3)
$\includegraphics {graph1.eps}$ (A) $\includegraphics {levels2.eps}$
(b): z=x³-2xy²-x²-y²
$\includegraphics {graph2.eps}$ (B) $\includegraphics {levels3.eps}$
(c): z=-(x²+y²) (C) $\includegraphics {levels1.eps}$
(d): z=x²-y² (D) $\includegraphics {levels4.eps}$

2.

(12%) The speed of sound C underwater (in m/s) depends on both the temperature T (in degrees Celsius) and the depth D (in meters). An approximate equation for the speed of sound is:

C=1449.2 + 4.6 T -0.055 T²+0.00029 T³ +0.016D.

(a): Find an expression for the rate of change of the sound speed with temperature. What do we call this rate of change?
(b): Find an expression for the rate of change of sound speed with depth. What do we call this rate of change?

3.

(20%) You have been commissioned to design a water trough with two ends which are sheet metal right triangles connected by sheet metal rectangular pieces attached to the two legs of the triangle, as shown. (See picture.) The top of the trough is left open. The trough should hold 1000 $\text{cm}^3$ of water, and use as little sheet metal as possible.

(a): What should the dimensions of the trough be? (Use one of the techniques discussed in class.)
(b): Explain briefly how you know that the point you found above is a minimum and not a maximum.

4.

(20%) Find the minimum and maximum values of x+y+z given the constraint that x²+y²+z²=25. (Use one of the techniques discussed in class.)

5.

(20%) The radius r and altitude h of a right circular cylinder are increasing at rates of 0.01 cm/min and 0.02 cm/min, respectively. Use the Chain Rule to find the rate at which the volume is increasing at the time when r=4 cm and h=7 cm.

6.

(20%) Suppose you are swimming in a lake where the depth in meters under the point (x,y) is f(x,y)=300-2x²-3y². You are at the point (3,4).

(a): If you swim in the direction of the unit vector $\langle \frac{\sqrt{3}}{2}, -\frac{1}{2} \rangle$ , will the depth increase or decrease? How fast?
(b): Now you are cold and want to get out of the lake. In what direction should you swim in order for the depth to decrease most rapidly? (You may give a vector of any length.) How fast will the depth decrease in that direction?

7.

THIS IS AN EXTRA CREDIT PROBLEM WORTH AN ADDITIONAL 5%.

Explain the geometrical significance of finding the maximum and minimum values of f(x,y,z)=z subject to the constraints

$\begin{displaymath} (x-1)^2+y^2+z^2=1 \text{\quad and \quad} z=\frac{x^2}{4}+\frac{y^2}{4}.\end{displaymath}$

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The translation was initiated by Joshua Holden on 11/21/1999

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Up: Math 103 Home Page

Joshua Holden
11/21/1999

Math 103

Test 2 -- Stage 2

Due October 27, 1999

About this document ...