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Related Rates. Objective: To find the rate of change of one quantity knowing the rate of change of another quantity. Related Rates.

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Related Rates

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Related rates

Related Rates

Objective: To find the rate of change of one quantity knowing the rate of change of another quantity.


Related rates1

Related Rates

  • When looking at water draining from a cone, we notice that the radius and height of the water changes as the volume changes. If we are interested in the rate of change of the volume with respect to time, we need to take the derivative of the volume function.


Related rates2

Related Rates

  • This is called a related rates problem because the goal is to find an unknown rate of change by relating it to other variables whose values and whose rates of change at time t are known or can be found.


Example 1

Example 1

  • Suppose that x and y are differentiable functions of t and are related by the equation . Find

    at time t = 1 if x = 2 and at time t = 1.


Example 11

Example 1

  • Suppose that x and y are differentiable functions of t and are related by the equation . Find

    at time t = 1 if x = 2 and at time t = 1.


Example 2

Example 2

  • Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2ft/s. How fast is the area of the spill increasing when the radius of the spill is 60ft?


Example 21

Example 2

  • Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2ft/s. How fast is the area of the spill increasing when the radius of the spill is 60ft?

  • We are looking for dA/dt.

  • We know dr/dt = 2 ft/s.

  • We know that r = 60 ft.


Example 22

Example 2

  • Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2ft/s. How fast is the area of the spill increasing when the radius of the spill is 60ft?

  • We are looking for dA/dt.

  • We know dr/dt = 2 ft/s.

  • We know that r = 60 ft.


Example 23

Example 2

  • Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2ft/s. How fast is the area of the spill increasing when the radius of the spill is 60ft?

  • We are looking for dA/dt.

  • We know dr/dt = 2 ft/s.

  • We know that r = 60 ft.


Example 3

Example 3

  • A baseball diamond is a square whose sides are 90 ft long. Suppose that a player running from second base to third base has a speed of 30 ft/s at the instant when s/he is 20 ft from third base. At what rate is the player’s distance from home plate changing at that time?


Example 31

Example 3

  • t = number of seconds since the player left second base

  • x = distance in feet from the player to third base

  • y = distance in feet from the player to home plate


Example 32

Example 3

  • We need to take the derivative to find the answer.


Example 33

Example 3

  • We need to take the derivative to find the answer.

  • We need to solve for y at the instant that x = 20


Example 34

Example 3

  • We need to take the derivative to find the answer.

  • We need to solve for y at the instant that x = 20


Example 4

Example 4

  • A camera is mounted at a point 3000 ft from the base of a rocket launching pad. If the rocket is rising vertically at 880 ft/s when it is 4000 ft above the launching pad, how fast must the camera elevation angle change at that instant to keep the camera aimed at the rocket?


Example 41

Example 4

  • A camera is mounted at a point 3000 ft from the base of a rocket launching pad. If the rocket is rising vertically at 880 ft/s when it is 4000 ft above the launching pad, how fast must the camera elevation angle change at that instant to keep the camera aimed at the rocket?

  • t = time in seconds

  • = angle of elevation

  • h = height of rocket


Example 42

Example 4

  • A camera is mounted at a point 3000 ft from the base of a rocket launching pad. If the rocket is rising vertically at 880 ft/s when it is 4000 ft above the launching pad, how fast must the camera elevation angle change at that instant to keep the camera aimed at the rocket?

  • t = time in seconds

  • = angle of elevation

  • h = height of rocket


Example 43

Example 4

  • We need to take the derivative to find the answer.


Example 44

Example 4

  • We need to find the when h = 4000.


Example 5

Example 5

  • Suppose that liquid is to be cleared of sediment by allowing it to drain through a conical filter that is

    16 cm high and has a radius of 4 cm at the top. Suppose also that the liquid is forced out of the cone at a constant rate of .

  • At what rate is the depth of the liquid changing at the instant when the liquid in the cone is 8 cm deep?


Example 51

Example 5

  • t = time elapsed

  • V = volume of liquid

  • h = depth of liquid

  • r = radius of liquid


Example 52

Example 5

  • The problem here is that we don’t know anything about r or dr/dt. We need to make a substitution to get rid of the r and put everything in terms of h.


Example 53

Example 5

  • The problem here is that we don’t know anything about r or dr/dt. We need to make a substitution to get rid of the r and put everything in terms of h.

  • We will use our knowledge of similar triangles to solve for r in terms of h.


Example 54

Example 5

  • We now need to find the derivative to solve for dh/dt.


Homework

Homework

  • Pages 221-223

  • 1-9 odd

  • 13-21 odd

  • 25,27


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