The fuel gauge of a truck driver friend of yours has quit working. His fuel tank is cylindrical like one shown below. Obviously he is worried about running out of fuel so he uses a stick he has in his cab which is about the same depth as his tank. He finds the fuel is about halfway up the stick. After driving for about 2 more hours, he decides to check again. Now the fuel is about ¼ of the way up the stick. He has about 1 and ½ hours until he reaches a neighboring state with much cheaper fuel prices. Your friend calls you up (being a calculus student and all…) and asks you whether he should try to make it to the neighboring state. Should he? Why or why not?

Lesson 2-6Related Rates

Finding Related Rates

If water is being pumped at , how fast is the water level

rising when h = 1 foot.

min2

3ft

Think of some quantities that we might use to solve this problem.

Solving Related Rates

1) Identify given and to be determined quantities

2) Write an equation relating these quantities

3) Differentiate

4) Substitute and solve

Finding Related Rates

If water is being pumped at , how fast is the water level

rising when h = 1 foot.

min2

3ft

dt

dh

Given: The change in Volume

(cubed)

dt

dVmin2

3ft

122

1hbVwater

What is it asking? How is the height changing w.r.t time.

Relevant equation?

Finding Related Rates

dt

dV

2) Differentiate implicitly w.r.t time (t) to obtain the related-rate.

]2**[3

)3(

2

2

dt

drrh

dt

dhr

pi

dt

dV

hrpi

dt

d

dt

dV

hrpi

Vwater2

3

1) What two variables do you want to compare?

Suppose you want to relate the change in volume w.r.t to the change in time.

Interpret this derivative. As the volume changes w.r.t time, the height and radius change as well.

Finding Related Rates

dt

drand

dt

dh

dt

dV,,

Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time.

a)How are the following related?

b) At a certain instant, the height is 6 in and increasing gat 1 in/s, while the radius is 10 in and decreasing at 1 in/s. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant.

Finding Related Rates

,32 xySuppose x and y are both differentiable functions of t and are related by the equation

Find when x=1, given that when x=1dt

dy2

dt

dx

Finding Related Rates

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?

What shape do we have?

What formula do we need?

What are we comparing?

What is given? (rates)

What are they asking?

Example

Example 4

Dx/dt is the velocity of the airplane…how fast is the airplane moving away from the antenna. What are the only two things that are changing? Your s and your x. Because your y is constant.

What are they asking? How fast your x is changing w.r.t time, horizontal distance? Find horizontal distance using Pythagorean theorem.What is given? ds/dtWhat do we find? dx/dt using equation x^2+6^2=s^2

Example

Find horizontal distance using Pythagorean theorem.What is given?What can we find?

Example

An 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 and the height is constantly at 6 miles, what is the speed of the plane?

Example 5

Find the rate of change in the angle of elevation of the camera shown at 10 seconds after lift-off.

Position function? What is the height at t=10?What rates are given? What rate do we have to find?What Trigonometric identity can we use to relate s and theta? What is changing?

Example 6

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 theta=pi/3.

What is a complete revolution? How do we obtain d0/dt?What rates are given? What rate do we have to find?What is the law of cosines?