4.6 Related Rates Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates.

4.6 Related Rates

• Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates.

Finding Related Rate Equationsa) Assume that the radius r of a sphere is a

differentiable function of t and let V be the volume of the sphere. Find an equation that relates dV/dt and dr/dt.

Finding Related Rate Equationsb.) Assume that the radius r and height h of a cone are

differentiable functions of t and let V be the volume of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.

Solution Strategy

A Rising Balloon• A hot-air balloon rising straight up from a level field is tracked

by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 radians per minute. How fast is the balloon rising at that moment?

A Rising Balloon• Step 1: Let h be the height of the balloon and

let theta be the elevation angle.– We seek: dh/dt– We know:

• Step 2: We draw a picture (Figure 4.55). We label the horizontal distance “500 ft” because it does not change over time. We label the height “h” and the angle of elevation “ .” Notice that we do not label the angle “π/4.”

/ 0.14 / mind dt rad

A Rising Balloon• Step 3:

• Step 4: Differentiate implicitly:

tan500

h

500 tanh

( ) (500 tan )d dh

dt dt

2500secdh d

dt dt

A Rising Balloon• Step 5: let

• Step 6: At the moment in question, the balloon is rising at the rate of 140 ft/min.

2500sec 0.144

dh

dt

/ 0.14, 4.d dt

2

500 2 0.14 140

A Highway Chase• 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. When the cruiser is 0.6 miles north of the intersection and the car is 0.8 miles to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of the measurement, what is the speed of the car?

A Highway Chase• Step 1: Let x be the distance of the speeding

car from the intersection, let y be the distance of the police cruiser from the intersection, and let z be the distance between the car and the cruiser. Distances x and z are increasing, but distance y is decreasing; so, dy/dt is negative.– We seek: dx/dt– We know: dz/dt = 20 mph and dy/dt = -60 mph.

• Step 2: A sketch (Figure 4.56) shows that x, y, and z form three sides of a right triangle.

A Highway Chase• Step 3:We need to relate those three variables,

so we use the Pythagorean Theorem:

x² + y² = z²• Step 4: Differentiate implicitly:

2 2 2dx dy dzx y zdt dt dt

dx dy dzx y zdt dt dt

A Highway Chase• Step 5: Substitute.

• Step 6: At the moment in question, the car’s speed is 70 mph.

0.8 36 1 20dx

dt

2 2(0.8) (0.6)( 60) 0.8 0.6 20

dx

dt

70dx

dt

Filling a Conical Tank• 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 radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

Filling a Conical Tank• Let V be the volume, r the radius, and h the height of

the cone of water.• We seek: dh/dt• We know: dV/dt = 9 ft3 / min.• The volume of the cone of water is:

• This formula also involves the variable r, whose rate of change is not given.

• We need to either find dr/dt or eliminate r from the equation, which we can do by using similar triangles.

21

3V r h

Filling a Conical Tank• We relate r and h:

• Therefore,

• Differentiate with respect to t:

Filling a Conical Tank• Let h = 6 and dV/dt = 9; then solve for dh/dt.

• At the moment in question, the water level is rising at 0.32 ft/min.

Filling a Conical Tank

• See Solution 2 on p. 249 in textbook.

Simulating Related Motion

• Parametric mode on a graphing calculator can be used to simulate the motion of moving objects when the motion of each can be expressed as a function of time.

• In a classic related rate problem, the top end of a ladder slides vertically down a wall as the bottom end is pulled horizontally away from the wall at a steady rate.

More Practice!!!!!

• Homework - Textbook p. 251 - 252 #1 – 17 odd.