Related Rates Problems: Applying the Chain Rule to Measure Change

1. Sec 3.4 Related Rates Problems – An Application of the Chain Rule

2. Derivative as Rate of Change Recall: If then 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 measures how fast y is changing with time.

3. Examples (1) If P denotes the population of a city, then measures how fast this population is changing with time. (2) If s denotes the distance between two cars, then 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 measures how fast the amount of water in that container is changing with time.

4. 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.

5. 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.

6. 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

7. 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.

8. 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.

9. Still One More Example Situation: 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.

10. 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.

11. 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.

12. 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).

13. Example 1: Solution This 50 cm3s-1 is the rate of change of volume of air inside the balloon. 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. If we let V denote the volume of air inside the balloon, … w … then dV/dt is 50. We want the rate of change of the diameter of the balloon. If we let w denote the diameter of the balloon, … … then we want to know dw/dt.

14. Example 2: Solution This 0.1 ms-1 measures the speed of the top of the ladder. 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. If we let y denote the vertical distance between the floor and the top of the ladder, … y 5 … then dy/dt is – 0.1 x If we define x as shown, … We want to find the speed of the bottom of the ladder. … then we want to know dx/dt.

15. Method of Solution: A Summary • Read and visualize what event is happening, and how the event occurs over time. • Identify what rate(s) of change is/are given, and what rate of change we are to find. • Write these rates as the derivativesof suitably chosen variables (with respect to time). • Find a relationship between these variables. • Obtain a relationship between the rates by (implicit) differentiation. • Find the unknown rate from the given rate(s).

16. Remarks • 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 …” • 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. • 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:

17. Example 3: Solution This is the speed of the man. 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. … then dx/dt is 0.2 … 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. If we call this x, … If y denotes the length of his shadow, …

18. Example 4: Solution 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. speed of the airplane If we call this distance x, … … then dx/dt is 400. Extra information we need: If we call the distance from the plane to the radar station y, … (1) The climbing angle, say, 30o y (2) The elevation of the airplane when it is directly above the radar station, say, 5000 meter … then we want to find dy/dt. (3) The position of the airplane

19. Example 5: Solution Situation: 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. Call this distance x Call this distance y z Call 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.

20. Example 6: Solution Top diameter, say, 100 cm. 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. Height, say, 150 cm. V h Rate of change of volume V of waterin container. dV/dtis –100. If we denote the depth of water in the container by h, … … we want to find dh/dt. Extra information needed: Dimensions of the (conical) container.

21. Example 7: Solution This is velocity … at a particular moment, say, 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. 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 whenhis 1000. Need what extra information ? If we use the angle to indicate the direction in question, … … then we want to find at that instant when h is 1000.

22. 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). … at the instant when the roller coaster is at the point, say (2,–4). dx/dt is 3 … … we want to find dy/dt … … at that particular instant.

23. Method of Solution: A Summary • Read and visualize what event is happening, and how the event occurs over time. • Identify what rate(s) of change is/are given, and what rate of change we are to find. • Write these rates as the derivativesof suitably chosen variables (with respect to time). • Find a relationship between these variables. • Obtain a relationship between the rates by (implicit) differentiation. • Find the unknown rate from the given rate(s).