Lab 5: Related Rates
In this lab you will do several problems that will help you explore and apply the concept of related rates. Do
the problems neatly in pencil on a separate piece of paper and staple your pages. Clearly lay out your work
using proper notation and write all final answers as sentences unless otherwise specified. Problems with just
an answer and no work will not receive credit. You are encouraged to work groups of 2 to 4 people and hand
in one write-up. This lab is worth 40 points. This is due next Friday, May 23.
1. (15 pts)
A train is traveling at 0.8 miles/min along a long straight track, moving as shown in the
diagram below. A movie camera, 0.5 miles away from the track, is focused on the train.
a) Express z as a function of x. (You don’t need to use a sentence.)
b) How fast is the distance from the camera to the train increasing when the train is 1 mile from the
camera?
c) How fast is the camera rotating at that moment? (Use radians/min.)
2. (15 pts)
A water tank is in the shape of an inverted cone with depth 10 meters and top radius 8
meters. Water is flowing into the tank at 0.1 cubic meters per minute but leaking out at a rate of
cubic meters per minute, where is the depth of the water in meters. (You don’t need to use
a sentence for part a or b.)
a) Find two expressions for . To find one of them, use the cone volume equation and the concept of
related rates (this expression should involve both and but not ). To find the other, use the two
pieces of information about water flow given in the problem.
b) Set the two expressions equal to each other and solve for .
c) When is positive? That is, what has to be true about ? You may use a calculator to help you
determine this.
d) Will the tank ever overflow? Explain, using your result from (c).
3. (10 pts)
There are two long parallel roads that run west-east; one is a gravel road, and the other
is a highway. They are 200 feet apart. Tom is driving 15 mph on the gravel road, going west. Jerry is
driving 60 mph on the highway, going east.
a) Convert their speeds to feet per second, showing your work. There are 5280 feet in a mile. (You
don’t need to use a sentence.)
b) How fast is the distance between Tom and Jerry changing 5 seconds after they pass one another?