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Calculus Related Rates

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Calculus Related Rates. In algebra we study relationships among variables. Solving Related Rates Equations. The volume of a sphere is related to its radius The sides of a right triangle are related by Pythagorean Theorem The angles in a right triangle are related to the sides.

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### CalculusRelated Rates

In algebra we study relationships among variables

Solving Related Rates Equations

• The volume of a sphere is related to its radius
• The sides of a right triangle are related by Pythagorean Theorem
• The angles in a right triangle are related to the sides.

In calculus we study relationships

between the rates of change of variables.

For example, how is the rate of change of the radius of a sphere related to the rate of change of the volume of that sphere?

Consider a sphere of radius 10cm.

If the radius changes 0.1cm/sec (a very small amount) how much does the volume change?

Figure 2.43: The balloon in Example 3.

Solving Related Rates Equations Ex. 2

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. The angle of elevation is increasing at the rate of 0.14 rad/min. How fast is the balloon rising when the angle of elevation is is /4?

Given:

Find:

Ex 2 con’t’d

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. At the moment the range finder’s elevation angle is /4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

What equation has all the knowns and unknowns in it?

Now take the derivative, implicitly

Figure 2.44: Figure for Example 4.

Ex 3

A police cruiser, approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. The cruiser is moving at 60 mph and the police determine with radar that the distance between them is increasing at 20 mph. When the cruiser is .6 mi. north of the intersection and the car is .8 mi to the east, what is the speed of the car?

Figure 2.44: Figure for Example 4.

Ex 3 con’t’d

A police cruiser, approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. The cruiser is moving at 60 mph and the police determine with radar that the distance between them is increasing at 20 mph. When the cruiser is .6 mi. north of the intersection and the car is .8 mi to the east, what is the speed of the car?

Given:

Find:

s

.6

.8

Ex 3 con’t’d

What equation has all the knowns and unknowns in it?

Now take the derivative, implicitly

Given:

Now sub in the given information

What is “S”?

, s = 1

Use Pythagorean Th.

solve

Ex 4

Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping? The radius is 15 cm.

What equation has all the knowns and unknowns in it?

(We need a formula to relate V and h.

(Since the radius is not changing, we treat it like a constant

(r is a constant.)

Ex 4

Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping? r = 15 cm

NOTE, the problem must be done in same units of measure

Ex 5

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep?

Given:

Find:

r

h

Ex 5

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep?

What equation has all the knowns and unknowns in it?

Given:

If we differentiate implicitly, we will have three rates, dV/dt, dh/dt, and dr/dt .

We cannot determine dr/dt. So we have to eliminate r in the equation.

Sub in for r in the volume formula

Ex 5

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep?

Given:

Solve for dh/dt

Find:

Truck Problem

Truck A travels east at 40 mi/hr.

Truck B travels north at 30 mi/hr.

Ex 6

How fast is the distance between the trucks changing 6 minutes later?

What equation has all the knowns and unknowns in it?

B

z

Now take the derivative, implicitly

y

x

A

Now sub in the given information

Ex 6

We must find x, y, and z at the specific time

Note 6 min = 1/10 hr.

B

Eastbound

Northbound

A

Truck Problem

Truck A travels east at 40 mi/hr.

Truck B travels north at 30 mi/hr.

Ex 6

How fast is the distance between the trucks changing 6 minutes later?

B

A

p

Solving Related Rates Equations
• Read the problem at least three times and draw a picture
• Identify all the given quantities and the quantities to be found (these are usually rates.)
• Draw a sketch and label, using unknowns when necessary.
• Write an equation (formula) that relates the variables.
• Assume all variables are functions of time and differentiate wrt time using the chain rule. The result is called the related rates equation.
• Substitute the known values into the related rates equation and solve for the unknown rate.