1. 2.8 Related Rates

2. Formulas You May Need To Know

3. Related rate problems are differentiated with
respect to time. So, every variable, except t is
differentiated implicitly.
Ex. Two rates that are related.
Given y = x2 + 3, find dy/dt when x = 1, given
that dx/dt = 2.
y = x2 + 3
Now, when x = 1 and dx/dt = 2, we
have

4. Assign symbols to all given quantities and
Procedure For Solving
Related Rate Problems
Assign symbols to all given quantities and
quantities to be determined. Make a sketch
and label the quantities if feasible.
Write an equation involving the variables
whose rates of change either are given or are
to be determined.
Using the Chain Rule, implicitly differentiate
both sides of the equation with respect to t.
Substitute into the resulting equation all known
values for the variables and their rates of change.
Solve for the required rate of change.

5. First, a review problem:
Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how much does the volume change?
The volume would change by approximately

6. Now, suppose that the radius is changing at an instantaneous rate of 0
Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec.
(Possible if the sphere is a soap bubble or a balloon.)
The sphere is growing at a rate of
Note: This is an exact answer, not an approximation like
we got with the differential problems.

7. Water is draining from a cylindrical tank at 3 liters/second
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
Find
(We need a formula to relate V and h. )
(r is a constant.)

8. Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.

9. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
Truck Problem:
Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A

10. p Truck Problem: Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later? B A p

11. Ex. A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The radius
r of the outer ripple is increasing at a constant rate
of 1 foot per second. When this radius is 4 ft., what
rate is the total area A of the disturbed water
increasing.
Givens:
Given equation:
Differentiate:

12. An inflating balloon
Air is being pumped into a spherical balloon at the
rate of 4.5 in3 per second. Find the rate of change
of the radius when the radius is 2 inches.
Given:
r = 2 in.
Equation:
Diff.
& Solve:

13. The velocity of an airplane tracked by radar
An airplane is flying at an elevation of 6 miles on a flight
path that will take it directly over a radar tracking station.
Let s represent the distance (in miles)between the radar
station and the plane. If s is decreasing at a rate of 400
miles per hour when s is 10 miles, what is the velocity of
the plane. s 6 x

14. Given:
Find:
Equation:
Solve:
x = s2
To find dx/dt, we
must first find x
when s = 10
Day 1

15. A fish is reeled in at a rate of 1 foot per second
from a bridge 15 ft. above the water. At what
rate is the angle between the line and the water
changing when there is 25 ft. of line out? x
15 ft.

16. Given:
Find:
Equation:
Solve:
x = 25 ft. h = 15 ft.

17. Ex. A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The radius
r of the outer ripple in increasing at a constant rate
of 1 foot per second. When this radius is 4 ft., what
rate is the total area A of the disturbed water
increasing.
An inflating balloon
Air is being pumped into a spherical balloon at the
rate of 4.5 in3 per minute. Find the rate of change
of the radius when the radius is 2 inches.

18. Example
Given
Find when x = 3 Note: we must differentiate implicitly with respect to t

19. Example Now substitute in the things we know Find other values we need
when x = 3, y2 = and y = 4

20. Example
Result

21. Guidelines for Related-Rate Problems
Identify given quantities, quantities to be determined
Make a sketch, label quantities
Write equation involving variables
Using Chain Rule, implicitly differentiate both sides of equation with respect to t
After step 3, substitute known values, solve for required rate of change

22. R1
Electricity R2
The combined electrical resistance R of R1 and R2 connected in parallel is given by
R1 and R2 are increasing at rates of 1 and 1.5 ohms per second respectively.
At what rate is R changing when R1 = 50 and R2 = 75?

23. Draining Water Tank Radius = 20, Height = 40
The flow rate = 80 gallons/min
What is the rate of change of the radius when the height = 12?

24. Draining Water Tank At this point in time the height is fixed
Differentiate implicitly with respect to t,
Substitute in known values
Solve for dr/dt

25. Assignment
Lesson 3.7
Page 187
Exercises 1 – 7 odd, 13 – 27 odd

26. Example #1
A ladder 10 feet long is resting against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 foot per second, how fast is the top of the ladder moving down when the bottom of the ladder is 8 feet from the wall?
First, draw the picture:

27. We have dx/dt is one foot per second. We want to find dy/dt.
X and y are related by the Pythagorean Thereom
Differentiate both sides of this equation with respect to t to get
When x = 8 ft, we have
Therefore
The top of the ladder is sliding down (because of the negative sign in the result) at a rate of 4/3 feet per second.

28. Example #2
A man 6 ft tall walks with a speed of 8 ft per second away from a street light atop an 8 foot pole. How fast is the tip of his shadow moving along the ground when he is 100 feet from the light pole.
18 ft
6 ft
z - x x z

29. Let x be the man’s distance from the pole and z be the distance of the tip of his shadow from the base of the pole.
Even though x and z are functions of t, we do not attempt to obtain implicit formulas for either.
We are given that dx/dt = 8 (ft/sec), and we want to find dz/dt when x = 100 (ft).
We equate ratios of corresponding sides of the two similar triangles and find that z/18 = (z-x)/6
Thus 2z = 3x

30. Implicit differentiation now gives 2 dz/dt = 3 dx/dt
We substitute dx/dt = 8 and find that
(dz/dt = 3/2) * (dx/dt = 3/2) * (8) = 12
So the tip of the man’s shadow is moving at 12 ft per second.

31. Try Me!
A ladder 25 ft long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at 3 ft/sec, how fast is the top of the ladder sliding down the wall, when the bottom is 15 ft from the wall?

32. Solution
t = the number of seconds in time that has elapsed since the ladder started to slide down the wall.
y = the number of feet in distance from the ground to the top of the ladder at t seconds.
x = the number of feet in the distance from the bottom of the ladder to the wall at t seconds.

33. Because the bottom of the ladder is pulled horizontally away from the wall at 3 ft/sec, dx/dt = 3. We wish to find dy/dt when x = 15.
From the Pythagorean Thereom, we have y^2 = 625 – x^2
Because x and y are functions of t, we differentiate both sides of equation one with respect to t and obtain 2y dy/dt = -2x dx/dt giving us dy/dt = -x/y dx/dt

34. When x = 15, it follows from equation one that y = 20.
Because dx/dt = 3, we get from equation two: dy/dt = (-15/20) * 3 = -9/4
Therefore, the top of the ladder is sliding down the wall at the rate of 2 ¼ ft/sec when the bottom is 15 ft from the wall.
The significance of the minus sign is that y is decreasing as t is increasing.

35. Was Your Answer Correct?