This document contains lecture slides on related rates from the Department of Mathematics at FSMT - UPSI. It begins with the learning outcome of solving problems in related rates. Then, it provides two examples - one involving a police radar gun calculating a driver's speed, and another involving calculating the rate of rise of water in a conical tank being filled. Both examples demonstrate setting up and solving related rates problems using differentiation.

1. Beginning Calculus
Applications of Di¤erentiation
- Related Rates -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 1 / 7

2. Related Rates
Learning Outcomes
Solve problems in related rates.
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 2 / 7

3. Related Rates
Example
Problem: Are you speeding? (For this problem, you are considered
speeding if exceeding 95ft/ sec)
30
x = 40
D = 50
Police
You
Given
dD
dt
= 80ft/ sec, the rate you are approaching (reado¤ from
the radar gun)
and known x2 + 302 = D2.
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 3 / 7

4. Related Rates
Problem - continue
d
dt
x2
+ 302
=
d
dt
D2
2x
dx
dt
= 2D
dD
dt
Plug-in the values for D and x.
2 (40)
dx
dt
= 2 (50) ( 80)
dx
dt
= 100ft/ sec
YES, you are speeding.
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 4 / 7

5. Related Rates
Example
Problem: A conical tank with top of radius 4ft and depth of 10ft, is
being …lled with water at 2ft3/ min . How fast is the water rising when it
is at depth of 5ft?
10
4
10
4
h
r
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 5 / 7

6. Related Rates
Example - continue
By similar traingle:
r
h
=
4
10
The volume of the cone:
V =
1
3
πr2
h,
dV
dt
= 2
The question is: Find
dh
dt
when h = 5
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 6 / 7

7. Related Rates
Example - continue
From
r
h
=
4
10
,
r =
2
5
h
Substitute and di¤erentiate: (
dV
dt
=
dV
dh
dh
dt
)
V =
1
3
π
2
5
h
2
h =
4
75
πh3
dV
dt
=
4
25
πh2 dh
dt
= 2
4
25
π (25)
dh
dt
= 2
)
dh
dt
=
1
2π
ft/ min
VillaRINO DoMath, FSMT-UPSI
(DA4) Applications of Di¤erentiation - Related Rates 7 / 7