Sec 3.4

Related Rates Problems –

An Application of the Chain Rule

Derivative as Rate of Change

Recall: If then ,xfy

xfdx

dy' the rate of change of y with respect

to x.

So, it measures how fast y is changing with respect to x.

In particular, if t stands for time, then

dt

dymeasures how fast y is changing with time.

Examples

(1) If P denotes the population of a city, then

dt

dPmeasures how fast this population is changing with time.

(2) If s denotes the distance between two cars, then

dt

ds measures how fast the distance between the two cars is changing with time.

(3) If V denotes the volume of water in some container, then

dt

dVmeasures how fast the amount of water in that container is changing with time.

Related Rates Problems

Situation: Some event occurs in time.

Given: How fast something is changing with time.

To find: How fast something else is changing with time.

Example

Situation: Air is being pumped into a (spherical) balloon.

Given: How fast air is being pumped into the balloon, say, 50 cm3 per second.

To find: How fast the diameter of the balloon is increasing with time.

Another Example

Situation: A ladder 5 meter in length, leaned against a wall, now starts to slide down.

Given: How fast the top of the ladder is sliding down, say, 0.1 meter per second.

To find: How fast the bottom of the ladder is sliding away from the wall.

5

Yet Another Example

Situation: A man is walking on a sidewalk.

Given: How fast the man is walking, say, 0.2 meter per second.

To find: How fast the size of his shadow on the ground is changing.

One more Example

Situation: An airplane is flying in the air, passing right over a radar station on the ground.

Given: How fast the airplane is flying, say, 400 ms-1 .To find: How fast its distance from the radar station is

changing.

Still One More ExampleSituation: Two ships are sailing.

Given: How fast (and in what direction) each ship is sailing, say 120 km/h and 160 km/h, respectively.

To find: How fast the distance between the ships is changing.

Here’s Another Example

Situation: Water is leaking from a container of the shape of an inverted cone.

Given: How fast water is leaking out, say, 100 cm3 per second.

To find: How fast the water level in the container is decreasing.

Here’s Yet Another Example

Situation: The space shuttle is being launched.

Given: How fast the space shuttle is rising some time after it is launched, say, 500 meter per second.

To find: How fast one needs to adjust the camera’s direction to keep the shuttle in sight.

One Last Example

Situation: A roller coaster moving along a track part of which is shaped like the parabola y = –x2.

Given: The horizontal speed at a certain point on the track, say, 3 meter per second.

To find: The vertical speed then (or there).

Example 1: Solution

Situation: Air is being pumped into a (spherical) balloon.

Given: How fast air is being pumped into the balloon, say, 50 cm3 per second.

To find: How fast the diameter of the balloon is increasing with time.

This 50 cm3s-1 is the rate of change of volume of air inside the balloon.

If we let V denote the volume of air inside the balloon, …

We want the rate of change of the diameter of the balloon.

If we let w denote the diameter of the balloon, …

w

… then dV/dt is 50.

… then we want to know dw/dt.

Example 2: Solution

Situation: A ladder 5 meter in length, leaned against a wall, now starts to slide down.

Given: How fast the top of the ladder is sliding down, say, 0.1 meter per second.

To find: How fast the bottom of the ladder is sliding away from the wall.

This 0.1 ms-1 measures the speed of the top of the ladder.

If we let y denote the vertical distance between the floor and the top of the ladder, …

… then dy/dt is – 0.1

y5

We want to find the speed of the bottom of the ladder.

If we define x as shown, …

x

… then we want to know dx/dt.

Method of Solution: A Summary

(1) Read and visualize what event is happening, and how the event occurs over time.

(2) Identify what rate(s) of change is/are given, and what rate of change we are to find.

(3) Write these rates as the derivatives of suitably chosen variables (with respect to time).

(4) Find a relationship between these variables.(5) Obtain a relationship between the rates by

(implicit) differentiation.(6) Find the unknown rate from the given rate(s).

Remarks

(1) The rate of change of a variable often depends on time. At different moments, the rate can be different. The question will specify the particular moment of time it is interested in. It is often phrased in the form “…when …”

(2) The equation you form to connect the variables must not rely on information that is valid for only a particular instant of the event. The equation has to be valid for the entire duration of the event.

(3) In Examples 3 onward, I had not displayed all necessary information that would allow us to solve them. Such information will be supplied as we discuss these examples now:

Example 3: Solution

Situation: A man is walking on a sidewalk.

Given: How fast the man is walking, say, 0.2 meter per second.

To find: How fast the size of his shadow on the ground is changing.

This is the speed of the man.

If we call this x, …

… then dx/dt is 0.2

If y denotes the length of his shadow, …

… then we want to find dy/dt .

Extra information we need (always given in question):

(1) Length of the lamppost, say, 5 m;

(2) Height of the man, say, 1.8 m.

Situation: An airplane is flying in the air, passing right over a radar station on the ground.

Given: How fast the airplane is flying, say, 400 ms-1 .To find: How fast its distance from the radar station is

changing.

Example 4: Solution

speed of the airplane

If we call this distance x, …… then dx/dt is 400.

If we call the distance from the plane to the radar station y, …

y

… then we want to find dy/dt .

Extra information we need:

(1) The climbing angle, say, 30o (2) The elevation of the airplane when it is directly above the radar station, say, 5000 meter(3) The position of the airplane

Example 5: SolutionSituation: Two ships are sailing.

Given: How fast (and in what direction) each ship is sailing, say 120 km/h and 160 km/h, respectively.

To find: How fast the distance between the ships is changing.

x

y

z

Call this distance

Call this distance

say. 120, is dt

dx

160. is dt

dy

dt

dzCall this distance

We want to know

Extra information we need:

(1) Original distance separating the ships, say, 200 km.

(2) Angle between their paths, say, 150o.

Example 6: Solution

Situation: Water is leaking from a container of the shape of an inverted cone.

Given: How fast water is leaking out, say, 100 cm3 per second.

To find: How fast the water level in the container is decreasing.

Rate of change of volume V of water in container.dV/dt is –100.

If we denote the depth of water in the container

by h, …

hV

… we want to find dh/dt.Extra information needed: Dimensions of the (conical) container.

Top diameter, say, 100 cm.

Height, say, 150 cm.

Example 7: SolutionSituation: The space shuttle is

being launched.

Given: How fast the space shuttle is rising some time after it is launched, say, 500 meter per second.

To find: How fast one needs to adjust the camera’s direction to keep the shuttle in sight.

This is velocity … at a particular moment, say,

when the shuttle reaches 1000 m above the ground.

If we denote the height of the shuttle above the ground

by h, …h

… then dh/dt is 500 when h is 1000.

If we use the angle to indicate the direction in question, …

… then we want to find at that instant when h is 1000.

dtd /

Need what extra information ?

Example 8: Solution

Situation: A roller coaster moving along a track part of which is shaped like the parabola y = –x2.

Given: The horizontal speed at a certain point on the track, say, 3 meter per second.

To find: The vertical speed then (or there).

dx/dt is 3 …

… at the instant when the roller coaster is at the point, say (2,–4).

… we want to find dy/dt … … at that particular instant.

Method of Solution: A Summary

(1) Read and visualize what event is happening, and how the event occurs over time.

(2) Identify what rate(s) of change is/are given, and what rate of change we are to find.

(3) Write these rates as the derivatives of suitably chosen variables (with respect to time).

(4) Find a relationship between these variables.(5) Obtain a relationship between the rates by

(implicit) differentiation.(6) Find the unknown rate from the given rate(s).