1. MATH 1910
Chapter 2
Section 6
Related Rates

2. Objectives Find a related rate.
Use related rates to solve real-life problems.

3. Finding Related Rates
You have seen how the Chain Rule can be used to find dy/dx implicitly.
Another important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time.

4. Finding Related Rates
For example, when water is drained out of a conical tank (see Figure 2.33), the volume V, the radius r,
and the height h of the water level are
all functions of time t.
Knowing that these variables are related
by the equation
Figure 2.33

5. Finding Related Rates
you can differentiate implicitly with respect to t to obtain the related-rate equation
From this equation, you can see that the rate of change of
V is related to the rates of change of both h and r.

6. Example 1 – Two Rates That Are Related
Suppose x and y are both differentiable functions of t and are related by the equation y = x2 + 3.
Find dy/dt when x = 1, given that dx/dt = 2 when x = 1.
Solution:
Using the Chain Rule, you can differentiate both sides of the equation with respect to t.
When x = 1 and dx/dt = 2, you have

7. Problem Solving with Related Rates
In Example 1, you were given an equation that related the variables x and y and were asked to find the rate of change when x = 1.
In the next example, you must create a mathematical model from a verbal description.

8. Example 2 – Ripples in a Pond
A pebble is dropped in a calm pond, causing ripples in the from of concentric circles. The radius r of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

9. Example 2 – Solution The variables r and A are related by
The rate of change of the radius is r = dr / dt = 1.
With this information, you can proceed as in Example 1.

10. Example 2 – Solution
cont’d
When the radius is 4 feet, the area is changing at a rate of 8π feet per second.

11. Problem Solving with Related Rates
cont’d

12. Problem Solving with Related Rates
cont’d
The table below lists examples of mathematical models involving rates of change.

13. Example 3
A ladder 10ft 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 top of the ladder sliding down the wall when the bottom of the ladder is 6ft from the wall.
𝑥 2 + 𝑦 2 =100
2𝑥 𝑑𝑥 𝑑𝑡 +2𝑦 𝑑𝑦 𝑑𝑡 =0
𝑑𝑦 𝑑𝑡 =− 𝑥 𝑦 𝑑𝑥 𝑑𝑡
Substitute in what is given
𝑑𝑦 𝑑𝑡 =− =− 3 4 𝑓𝑡/𝑠

14. Example 4
A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2 𝑚 3 /𝑚𝑖𝑛, find the rate at which the water level is rising.
𝑟= ℎ 2
𝑑𝑉 𝑑𝑡 =2 𝑚 3 /𝑚𝑖𝑛
𝑉= 1 3 𝜋 𝑟 2 ℎ
𝑟 ℎ = 2 4
ℎ=3
𝑑ℎ 𝑑𝑡 =?
𝑉= 1 3 𝜋 ℎ ℎ
𝑉= 𝜋 12 ℎ 3
𝑑𝑉 𝑑𝑡 = 3𝜋 12 ℎ 2 𝑑ℎ 𝑑𝑡
𝑑ℎ 𝑑𝑡 = 4 𝜋 ℎ 2 𝑑𝑉 𝑑𝑡
Therefore
Substitute Givens
𝑑ℎ 𝑑𝑡 = 8 9𝜋 =0.28𝑚/𝑚𝑖𝑛

15. Example 5
A man walks along a straight path at a speed of 4ft/s. A searchlight is located on the ground 20ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15ft from the point on the path closest to the searchlight?
𝑑𝑥 𝑑𝑡 =4𝑓𝑡/𝑠
𝑑𝜃 𝑑𝑡 =?
𝑥=15
𝑥 20 =𝑡𝑎𝑛𝜃
𝑥=20𝑡𝑎𝑛𝜃
𝑑𝑥 𝑑𝑡 =20 𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡
𝑑𝜃 𝑑𝑡 = 𝑐𝑜𝑠 2 𝜃 𝑑𝑥 𝑑𝑡 = 𝑐𝑜𝑠 2 𝜃 4 = 1 5 𝑐𝑜𝑠 2 𝜃
𝑥=15, 𝑙𝑒𝑛𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑒𝑎𝑚 𝑖𝑠 25
𝑠𝑜 𝑐𝑜𝑠𝜃= 4 5
𝑑𝜃 𝑑𝑡 = = =0.128 𝑟𝑎𝑑/𝑠
Substitute in

16. Example 6
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 𝑐𝑚 2 /𝑠. How fast is the radius of the balloon increasing when the diameter is 50cm?
𝐺𝑖𝑣𝑒𝑛: 𝑑𝑉 𝑑𝑡 =100 𝑐𝑚 2 /𝑠
𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=50𝑐𝑚
𝑠𝑜 𝑟=25𝑐𝑚
𝑑𝑟 𝑑𝑡 =?
𝑉= 4 3 𝜋 𝑟 3
𝑑𝑉 𝑑𝑡 = 12 3 𝜋 𝑟 2 𝑑𝑟 𝑑𝑡
𝑑𝑟 𝑑𝑡 = 1 4𝜋 𝑟 2 𝑑𝑉 𝑑𝑡
Substitute
Givens
𝑑𝑟 𝑑𝑡 = 1 4𝜋 = 1 25𝜋