MATH 100 V1A
November 21st – Practice problems
Hints and Solutions
1. Consider an aircraft flying north at 600 km/hr, at an altitude of 4km, passing directly
overhead a car driving east at 100 km/hr. Determine how fast the distance between
them is changing one hour after the aircraft passes overhead the car. You do not have
to simplify your answer.
Hint: If x denotes the distance the car has travelled east and y denotes the distance
the plane has flown north, show that the distance between the car and the plane is
given by
p
D = x2 + y 2 + 16.
Use this to show that the distance between them is changing at a rate of
km/hr one hour after the aircraft passes overhead the car.
2
2
√ 100 +600
1002 +6002 +16
2. As a woman walks away from a street lamp, her shadow lengthens. Prove that it does
so at a rate which depends on her speed but not on her distance from the lamppost.
Solution: Let H denote the height of the lamppost, h the height of the woman, y
the distance between the woman and the lamppost and x the length of the woman’s
shadow. Then the “large” triangle formed by the top of the lamppost, the bottom of
the lamppost and the end of her shadow is similar to the “small” triangle formed by
the top of her head, the bottom of her feet and the end of her shadow. This means
that
x
x+y
=
.
h
H
h dy
h
So, x = H−h
y, and so dx
= H−h
. Since dy
is the woman’s speed, we conclude that
dt
dt
dt
the rate at which her shadow lengthens depends on the woman’s speed, but not on her
distance from the lamppost (since y itself does not appear in our equation for dx
).
dt