Ch 4.6 Related RatesGraphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy

Differentiation With Respect to Different Variables

The volume of a cone can be expressed as:2r h

V = 3

a) Find dV/dt if V, r, and h are all functions of time.

b) Find dV/dt if r is constant so that V and h are functions of time.

c) Find dV/dt if V is constant so that only r and h are functions of time.

Differentiation With Respect to Different Variables

2dV r dh =

dt 3 dt

2dV dh dr = r + 2rh

dt 3 dt dt

The volume of a cone can be expressed as:2r h

V = 3

a) Find dV/dt if V, r, and h are all functions of time.

b) Find dV/dt if r is constant so that V and h are functions of time.

c) Find dV/dt if V is constant so that r and h are functions of

time.2 dh dr

0 = r + 2rh3 dt dt

Related Rate Example

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 .14 radians/min. How fast is the balloon rising at that moment?

Related Rate Example

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 .14 radians/min. How fast is the balloon rising at that moment?

4

h

500 ftR.F.

2

2

d dhradAt = , = .14 , Find 4 mindt dth

1. tan = h = 500 tan 500

dh d2. = 500 sec

dt dtdh

3. At = , = 500 sec .144 4dtdh ft = 1000 .14 = 140 secdt

Steps to Solving a Related Rate Problem

1. Identify the variable whose rate of change we want and the variables whose rate of change we have.

2. Draw a picture and label with values and variables.

3. Write an equation relating the variable wanted with known variables.

4. Differentiate implicitly with respect to time.

5. Substitute known values

6. Interpret the solution and reread the problem to make sure you’ve answered all the questions.

.6

.8

zy

x

Related Rate Example

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?

Related Rate Example

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?

h10 ft

r

5 ft2

2 33

2

dV dh = 9, r = 5 , Find at h = 6

dt dt1 r 1 h

1. V = r h, but = , so r = 3 h 2 2

1 h 1 h2. V = h = = h

3 2 3 4 12

dV dh3. = h

dt 4 dtdh

4. 9 = 364 dt

dh 1 ft = = .31831 secdt

Related Rate Example

A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec.

1. Show how the motion of the two ends of the ladder can be represented by parametric equations.

2. What values of t make sense in this problem.

3. Graph using a graphing calculator in simultaneous mode. State an appropriate viewing window. (Hint: hide the coordinate axis).

4. Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t = .5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground?

Related Rate Example

A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec.

1. Show how the motion of the two ends of the ladder can be represented by parametric equations.

2. What values of t make sense in this problem.

{ 0 < t < 13/3 }

1 1 1

222 2

x t = 3t, y t = 0 x t = distance from base of wall at time = t

x t = 0, y t = 13 - 3t = height of ladder at time = t

Related Rate Example

3. Graph using a graphing calculator in simultaneous mode. State an appropriate viewing window. (Hint: hide the coordinate axis). t: [0,4] x: [0,13] y: [0, 13]

4. Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t = .5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground?

2 2y + x = 169

dy x dx = -

dt y dt

dy dy dy dy.5 -.348, 1 -.712, 1.5 -1.107, 2 -1.561

dt dt dt dtdy ftWhen ladder hits ground, (4.3) - 24.05 secdt