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 Equation of a circle centered at the origin with radius of 2: –x 2 + y 2 = 4

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

What is a related rate?

TABLE OF CONTENTS  AREA AND VOLUME AREA AND VOLUME  PYTHAGOREAN THEOREM AND SIMILARITY PYTHAGOREAN THEOREM AND SIMILARITY  TRIGONOMETRY TRIGONOMETRY  MISCELLANEOUS EQUATIONS MISCELLANEOUS EQUATIONS

AREA AND VOLUME RELATED RATES

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?

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

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?

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

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.

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

Example 4 Gravel is being dumped from a conveyor belt at a rate of 30 ft 3 /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?

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

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

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.

Ex 5: Answer Volume of a Cone: Given: Find: when h = 3 cm 3 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

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

PYTHAGOREAN THEOREM AND SIMILARITY

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.

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

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

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?

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

Ex 7: Balloon and Bicycle - solution s x y

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.

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

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

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

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.

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

Ex 9: Answer (con’t) xs 15 6 Table of contents

RELATED RATES WITH TRIGONOMETRY

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?

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

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

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 3 rd base?

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

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

MISCELLANEOUS EQUATIONS

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?

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