1 / 42

RELATED RATES

RELATED RATES. DERIVATIVES WITH RESPECT TO TIME. How do you take the derivative with respect to time when “time” is not a variable in the equation?. Consider a circle that is growing on the coordinate plane: Growing Circle Animation

vbartlett
Download Presentation

RELATED RATES

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. RELATED RATES DERIVATIVES WITH RESPECT TO TIME

  2. How do you take the derivative with respect to time when “time” is not a variable in the equation? • Consider a circle that is growing on the coordinate plane: • Growing Circle Animation • Equation of a circle centered at the origin with radius of 2: • x2 + y2 = 4

  3. In each case find the derivative with respect to ‘t’. Then find dy/dt.

  4. What is a related rate?

  5. TABLE OF CONTENTS • AREA AND VOLUME • PYTHAGOREAN THEOREM AND SIMILARITY • TRIGONOMETRY • MISCELLANEOUS EQUATIONS

  6. AREA AND VOLUME RELATED RATES

  7. Example 1 Suppose a spherical balloon is inflated at the rate of 10 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches?

  8. Ex 1: Answer Volume of a Sphere: Given: Find: when r = 5 inches

  9. Example 2 A shrinking spherical balloon loses air at the rate of 1 cubic inch per minute. At what rate is its radius changing when the radius is (a) 2 inches? (b) 1 inch?

  10. Ex 2: Answer Volume of a Sphere: Given: Find: when a) r = 2 inches b) r = 1 inch

  11. Example 3 The area of a rectangle, whose length is twice its width, is increasing at the rate of Find the rate at which the length is increasing when the width is 5 cm.

  12. Ex 3: Answer Area of a rectangle: Given: l = 2w Find: when w = 5 cm l = 10 cm

  13. Example 4 • Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

  14. Ex 4: Answer Volume of a Cone: Given: d = h or 2r = h Find: when h = 10 ft Eliminate ‘r’ from the equation and simplify

  15. Ex 4: Answer (con’t) Take the derivative Substitute in the specific values and solve. Table of contents

  16. Example 5 An inverted conical container has a height of 9 cm and a diameter of 6 cm. It is leaking water at a rate of 1 cubic centimeter per minute. Find the rate at which the water level h is dropping when h equals 3cm.

  17. Ex 5: Answer 3 Volume of a Cone: Given: Find: when h = 3 cm 9 Since the base radius is 3 and the height of the cone is 9, the radius of the water level will always be 1/3 of the height of the water. That is r = 1/3h

  18. Ex 5: Answer (con’t) 3 Volume of a Cone: 9 Table of contents

  19. PYTHAGOREAN THEOREM AND SIMILARITY

  20. Example 6 A 13 meter long ladder leans against a a vertical wall. The base of the ladder is pulled away from the wall at a rate of 1 m/s. Find the rate at which the top of the ladder is falling when the base of the ladder is 5m away from the wall.

  21. Ex 6: Answer Given: Length of ladder – 13 m Find: when x = 5 m 13 y x Use Pythagorean Theorem to relate the sides of the triangle!

  22. Ex 6: Answer (con’t) By the Pythagorean Thm: 13 y x Find ‘y’ when x = 5 using Pythagorean Thm.

  23. Ex 7: A balloon and a bicycle A balloon is rising vertically above a level straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?

  24. Ex 7: Balloon and Bicycle - solution Given: rate of balloon rate of cyclist Find: when x = ? and y = ? Distance = rate * time s y x

  25. Ex 7: Balloon and Bicycle - solution s y x

  26. Ex 8: The airplane problem- A highway patrol plane flies 3 mi above a level, straight road at a steady pace 120 mi/h. The pilot sees an oncoming car and with radar determines that at the instant the line of sight distance from plane to car is 5 mi, the line of sight distance is decreasing at the rate of 160 mi/h. Find the car’s speed along the highway.

  27. Ex 8: Airplane - solution Given: rate of plane: when s=5: Find: rate of the car:

  28. Ex 8: Airplane – solution(con’t) p 3 p + x 3 s s 3 (x+p)

  29. Ex 8: Airplane – solution(con’t) s 3 (x+p)

  30. Example 9 A 6 foot-tall man is walking straight away from a 15 ft-high streetlight. At what rate is his shadow lengthening when he is 20 ft away from the streetlight if he is walking away from the light at a rate of 4 ft/sec.

  31. Ex 9: Answer Given: streetlight – 15 ft man – 6 ft Find: when x = 20 ft 15 6 x s Set up a proportion using the sides of the large triangle and the sides of the small triangle.

  32. Ex 9: Answer (con’t) 15 6 x s Table of contents

  33. RELATED RATES WITH TRIGONOMETRY

  34. Example 10 A ferris wheel with a radius of 25 ft is revolving at the rate of 10 radians per minute. How fast is a passenger rising when the passenger is 15 ft higher than the center of the ferris wheel?

  35. Ex 10: Answer Given: Radius – 25 ft Find: when y = 15 ft. 25 y 

  36. Ex 10: Answer Find cos  when y = 15 ft 25 y 

  37. Example 11 A baseball diamond is a square with sides 90 ft long. Suppose a baseball player is advancing from second to third base at a rate of 24 ft per second, and an umpire is standing on home plate. Let  be the angle between the third base line and the line of sight from the umpire to the runner. How fast is  changing when the runner is 30 ft from 3rd base?

  38. Ex 11: Answer Given: Side length – 90 ft. Find: when x = 30 ft. x 90 

  39. Ex 11: Answer (con’t) Solve equation for d/dt. Find cos  when x = 30: x 90  Table of contents

  40. MISCELLANEOUS EQUATIONS

  41. Example 12 An environmental study of a certain community indicates that there will be units of a harmful pollutant in the air when the population is p thousand. The population is currently 30,000 and is increasing at a rate of 2,000 per year. At what rate is the level of air pollution increasing?

  42. Ex 12: Answer Given: Find: when p =30thous/yr.

More Related