Math 1320 - Lab 13
Name:
uNiD:
Due: 12/12
READ BEFORE PROCEEDING: Today we will continue with some application problems in multivariate calculus (specifically partial derivatives). Don’t forget to write your reasoning (in COMPLETE and
CONCISE sentences) behind your approach to each problem so that someone who has taken Math 1310
would understand what you are doing. Present your work in such a way that someone without access to
the questions can read through your solutions and understand it. To achieve this, set-up the problem in
your own words and describe NEATLY and LEGIBLY the procedures you take in solving the problem.
Disjoint or unclear explanations may result in lost points.
1. The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance
R is slowly increasing as the resistor heats up.
(a) Use Ohm’s Law, V = IR, to find how the current I is changing at the moment when R = 400 Ω,
I = 0.08 A, dV /dt = −0.01 V/s, and dR/dt = 0.03 Ω/s.
(b) What can you infer from your solution in (a)?
(c) What is more responsible for the current loss, the battery wearing out or the resistor heating up?
Explain.
2. An alpine roller coaster is to be constructed along the side of a hill. The hill is modeled by the equation
z = 50 − x2 + 5xy − 2y 2 , where x, y and z are measured in meters. You are a member of the design team.
You are tasked with providing information about possible roller coaster track paths at a point (−1, 3, 16)
on the hill. You are asked to answer the following questions with adequate mathematical justification.
Assume the positive y-axis points north and the positive x-axis points east.
(a) In what direction is the steepest downward path? Why is this information important?
(b) In what direction is the path level? Why is this information important?
(c) If you angle the roller coaster track due north, will you start to ascend or descend?
(d) If you angle the roller coaster track southeast:
(i) Will you start to ascend or descend?
(ii) At what rate?
(iii) At what angle above or below the horizontal will the path in that direction begin?
(e) If the roller coaster track is toward the point (0, 5, 0) on the hill, will you start to ascend or descend?
At what rate?
3. A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a
rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 per day, the floor at a
rate of 1 unit/m2 per day, and the roof at a rate of 5 units/m2 per day. Each wall must be at least 30
m long, the height must be at least 4 m, and the volume must be exactly 4000 m3 .
(a) Derive a function that describes the heat loss of the building. What is the domain of this function?
(b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the
boundary of the domain.)
(c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls
were removed?