This document contains examples of solving related rates problems. The first example involves calculating the rate of increase of the radius of a spherical balloon as it inflates at a rate of 100 cm3/s. When the diameter is 50 cm, the radius is increasing at a rate of approximately 0.0127 cm/s. The document provides guidance on setting up related rates problems, including drawing diagrams, identifying known and unknown variables, relating variables through equations, and differentiating to solve for the unknown rate. Two additional examples involve calculating rates of change for a filling water tank and a sliding ladder.

1. Related 4.1 Rates

2. 2
Example 1 – Inflating a Balloon
Air is being pumped into a spherical balloon so that its
volume increases at a rate of 100 cm3/s. How fast is the
radius of the balloon increasing when the diameter is
50 cm?
Solution:
We start by identifying two things:
the given information:
the rate of increase of the volume of air is 100 cm3/s
and the unknown:
the rate of increase of the radius when the diameter
is 50 cm

3. cont’d
3
Example 1 – Solution
In order to express these quantities mathematically, we
introduce some suggestive notation:
Let V be the volume of the balloon and let r be its radius.
Key idea:
rates of change are derivatives
The rate of increase of the volume with respect to time is the
derivative dV/dt,
the rate of increase of the radius is dr /dt.

4. 4
Example 1 – Solution
Given: = 100 cm3/s
Unknown: when r = 25 cm
connect dV/dt and dr/dt  relate V and r
V = pr 3
cont’d

5. 5
Example 1 – Solution
Differentiate each side of this equation with respect to t.
For right side, we need to use the Chain Rule:
Now we solve for the unknown quantity:
cont’d

6. 6
Example 1 – Solution
If we put r = 25 and dV/dt = 100 in this equation, we obtain
The radius of the balloon is increasing at the rate of
1/(25p) ≈ 0.0127 cm/s.
cont’d

7. 7
Solving Related Rates - Process
1. Read the problem (several times)
2. Draw a diagram
3. Label variables
4. Identify givens
5. Identify what you are trying to find (usually a
derivative)
6. Relate variables  equation
 May need to eliminate varaibles
1. Differentiate with respect to time
2. Solve for unknown
3. Answer question, with units!!

8. Example – Eliminating a Variable
A water tank has the shape of an inverted circular cone with
a base radius of 2 m and height of 4 m. If water is being
pumped into the tank at a rate of 2 m3/min, find the rate at
which the water level is rising when the water is 3 m deep.
8

9. Example – Implicit Differentiation
A ladder 10 ft long rests against a vertical wall. If the bottom
of the ladder slides away from the wall at a rate of 1 ft/s,
how fast is the ladder sliding down the wall when the bottom
of the ladder is 6 ft from the wall?
9

10. 10
Example
A man walks along a straight path at a speed of 4 ft/s. A
searchlight is located on the ground 20 ft from the path and
is kept focused on the man. At what rate is the seachlight
rotating when the man is 15 ft from the point on the path
closest to the searchlight?