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RELATED RATES

RELATED RATES. DERIVATIVES WITH RESPECT TO TIME. Example 1. 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 2 inches?. Ex 1 : Answer. Volume of a Sphere: Given: Find: when r = 2 inches. Example 2.

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RELATED RATES

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  1. RELATED RATES DERIVATIVES WITH RESPECT TO TIME

  2. Example 1 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 2 inches?

  3. Ex 1: Answer Volume of a Sphere: Given: Find: when r = 2 inches

  4. Example 2 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.

  5. Ex 2: 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

  6. Ex 2: Answer (con’t) 3 Volume of a Cone: 9 Table of contents

  7. Ex 3: 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? How fast is the angle formed between the cyclist, the balloon and the ground changing at that same instant?

  8. Ex 3: Balloon and Bicycle - solution Given: rate of balloon rate of cyclist Find: when x = ? and y = ? Distance = rate * time s y x

  9. Ex 3: Balloon and Bicycle - solution s y x

  10. Ex 4: 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.

  11. Ex 4: Airplane - solution Given: rate of plane: when s=5: Find: rate of the car:

  12. Ex 8: Airplane – solution(con’t) p 3 p + x 5 s s 3 (x+p)

  13. Ex 4: Airplane – solution(con’t) s 3 (x+p)

  14. 2007 – Form B – Question 3 The wind chill is the temperature, in degrees Fahrenheit (oF), a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour (mph). If the air temperature is 32oF, then the wind chill is given by W(v) = 55.6 – 22.1v0.16and is valid for 5 ≤ v ≤ 60 . a) Find W’(20). Using correct units, explain the meaning of W’(20) in terms of the wind chill. b) Find the average rate of change of W over the interval . Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval . c) Over the time interval 0 ≤ t ≤ 4 hours, the air temperature is a constant 32oF. At time t = 0, the wind velocity is v = 20 mph. If the wind velocity increases at a constant rate of 5 mph, what is the rate of change of the wind chill with respect to time at t = 3 hours? Indicate units of measure.

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