1. Aim: How do we find related rates when we have more than two variables?
Do Now:
Find the points on the curve x2 + y2 = 2x +2y where the tangent is parallel to the x-axis.
(1, /2), (1, 1 – 21/2)

2. Related Rates involving TIME
Most quantities encountered in science or everyday life vary with time.
If two such quantities are related to each other by an equation, and if we know the rate at which one of them changes, then, by differentiating the equation with respect to time, we can find the rate at which the other quantity changes.
Implicit differentiation and the Chain Rule

3. Related Rates involving TIME
Suppose a particle starts from the origin and moves along the curve y = x2.
as it moves both x and y values change
Suppose:
the rate of change for x is ½ unit/sec.
How can we find the rate of change of y?
rate of change of y equals x at that point.

4. Related Rates involving TIME
Suppose that the diameter D and the area of a circle are differentiable functions of t. Write an equation that relates dA/dt and dD/dt.

5. Model Problem
A particle moves along the curve y = 3x2 – 6x so that the rate of change of the x coord. is
Find the rate of change of the y-coord. when the particle is at the origin.
@ x = 0
@ x = 0

6. Find the rate of change of the x-coord.
Model Problem
A particle moves along a circle 25 = x2 + y2. If the particle is at (-4, 3) and the y-coord. is increasing so that
Find the rate of change of the x-coord.
Take derivative of x2 + y2 = 25 with respect to t
Substitute and solve for dx/dt.

7. Model Verbal Problem Guidelines
Identify all given quantities and quantities to be determined. Make a sketch and label quantities.
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 time t.
After completing Step 3, substitute into the resulting equation all known values for the variable and their rates of change and then solve.

8. Related Rates involving TIME
volume in dependent on measurements of radius and height
If water were draining out of this cone, the volume V, the height h, and the radius r, of the water level would all be functions of time t.

9. The Draining Cone
implicitly differentiate the Volume formula in terms of time t.
this reinforces the fact that the rate of Volume change is related to the changes in h and r.

10. The Draining Cone
If the water height is changing at a rate of foot per minute and the radius is changing at the rate of -0.1 foot per minute, what is the rate of change in the volume when the radius r = 1 foot and the height h = 2 feet?
Chain Rule
= cubic feet per min.

11. When x = 1 and dx/dt = 2, you have
Model Problem
x and y are both differentiable functions of t and related by the equation y = x Find dy/dt when x = 1, given that dx/dt = 2 when x = 1.
y = x2 + 3
original equation
differentiate with respect to t
Chain Rule
When x = 1 and dx/dt = 2, you have

12. Aim: How do we find related rates when we have more than two variables?
Do Now:
Suppose that the diameter D and the area of a circle are differentiable functions of t. Write an equation that relates dA/dt and dD/dt.

13. Aim: How do we find related rates when we have more than two variables?
Do Now:
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 the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

14. Model Verbal Problem
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 the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
Area formula for circle
rate of change
what to find?

15. When r = 4, the area is changing at a rate of 8 sq.ft./sec.
Model Verbal Problem
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 the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
differentiate with respect to t
Chain Rule
Substitute
When r = 4, the area is changing at a rate of 8 sq.ft./sec.

16. Model Verbal Problem
Air is being pumped into a spherical balloon at a rate of 4.5 cubic inches per minute. Find the rate of change of the radius when the radius is 2 inches.
Volume formula for sphere
rate of change
what to find?

17. Model Problem (con’t)
Air is being pumped into a spherical balloon at a rate of 4.5 cubic inches per minute. Find the rate of change of the radius when the radius is 2 inches.
Volume formula for sphere
differentiate with respect to t
solve for dr/dt
substitute

18. Model Problem
A airplane is flying at an altitude of 6 mi. on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?
a2 + b2 = c2
rate of change
when s = 10, x = 8 miles x
6 mi. s
what to find? 10 8 8

19. Model Problem
A airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?
x = s2
Pythagorean Theorem
differentiate with respect to t x
6 mi. s
solve for dr/dt
substitute

20. Substitution & Evaluation
Model Problem
Find the rate of change in the angle of elevation of the camera at 10 seconds after liftoff.
 is the angle of elevation
s = 50t2
Position Equation
Velocity of rocket
when t = 10, s = 5000 feet
Substitution & Evaluation
what to find?

21. Model Problem
Find the rate of change in the angle of elevation of the camera at 10 seconds after liftoff.
differentiate with respect to t
substitute ds/st = 100t and solve for d/dt
substitute s, and t and simplify

22. Aim: How do we find related rates when we have more than two variables?
Do Now:
At a sand and grave plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?

23. Model Problem
In an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when  = /3. 7 3
what to find?
crank
connecting rod 
= /3 x
the velocity of a piston is related to the angle of the crankshaft. ?
How do we relate x to ?

24. differentiate with respect to t
Model Problem
In an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when  = /3.
How do we relate x to ?
Law of Cosines: b2 = a2 + c2 – 2ac cos  x a b 
72 = 32 + x2 – 2(3)(x) cos 
differentiate with respect to t
isolate dx/dt

25. Model Problem 72 = 32 + x2 – 2(3)(x) cos /3 49 = 9 + x2 – 6(x)(1/2)
In an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when  = /3.
72 = 32 + x2 – 2(3)(x) cos /3
solve for x by substituting  = /3
49 = 9 + x2 – 6(x)(1/2)
0 = x2 – 3x - 40
0 = (x – 8)(x + 5) x = 8
solve for dx/dt by substituting x = 8 and  = /3

26. Model Problem
A circular pool of water is expanding at the rate of 16π in2/sec. At what rate is the radius expanding when the radius is 4 inches?

27. Model Problem
A 25-foot ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of 15 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 7 feet from the ground?

28. Model Problem
A spherical balloon is expanding at a rate of 60π in3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 in.

29. Model Problem
Model Problem
An underground conical tank, standing on its vertex, is being filled with water at the rate of 18π ft3/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?
An underground conical tank, standing on its vertex, is being filled with water at the rate of 18π ft3/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?

30. Model Problem
A rocket is rising vertically at a rate of 5400 miles per hour. An observer on the ground is standing 20 miles from the rocket’s launch point. How fast (in radians per second) is the angle of elevation between the ground and the observer’s line of sight of the rocket increasing when the rocket is at an elevation of 40 miles?

31. Model Problem
A man 2 meters tall walks at the rate of 2 meters per second toward a streetlight that’s 5 meters above the ground. At what rate is the tip of his shadow moving?