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Related Rates

Related Rates. The Hoover Dam. Example of a Related Rate:. Oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s, how fast is the area of the spill increasing when the radius in 30m?.

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Related Rates

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  1. Related Rates

  2. The Hoover Dam

  3. Example of a Related Rate: Oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s, how fast is the area of the spill increasing when the radius in 30m?

  4. Step 1: Read the problem carefully. Step 2: Draw a picture to model the situation. Step 3:Identify variables of the known and the unknown. Some variables may be rates. Step 4: Write an equation relating the quantities. Step 5: Implicitly differentiate both sides of the equation with respect to time, t.

  5. Step 6: Substitute values into the derived equation. Step 7: Solve for the unknown. Step 8: Check your answers to see that they are reasonable. CAUTION: Be sure the units of measurement match throughout the problem. CAUTION: Be sure to include units of measurement in your answer.

  6. The table below lists examples of mathematical models involving rates of change. Let’s translate them into variable expressions:

  7. Geometry Formula Review h c r a r r b V = r2h C = 2 r A = r2 a2 + b2 = c2 V = 4/3r3 SA = 4 r2 30 x x/2√3 h h 60 r b x/2 A = 1/2 bh V = 1/3 r2h

  8. Let’s try: Oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s, how fast is the area of the spill increasing when the radius in 30m? What formula can I use? Substitute in what you know! How can I get dA/dtout of that formula? What are we trying to find? What variable can we assign this unknown? dA/dt = 2rdr/dt dA dt =? dA/dt = 2(30 m)(1 m/s) A = r2 dA/dt = 60 m2/s

  9. Your turn: A child throws a stone into a still pond causing a circular ripple to spread. If the radius increases at a constant rate of 1/2m/s, how fast is the area of the ripple increasing when the radius of the ripple is 20 m? Answer: 20 m2/s or 62.8 m2/s

  10. The process might get more involved. If a snowball (perfect sphere) melts so that its surface area decreases at a rate of 1 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm. We have to rewrite this formula so that it has a diameter instead of a radius… What variable can we use to define the unknown? What formula can we use? Can you finish from here? How can we get dd/dt out of this formula? What are we trying to find? r dd dt =? SA = 4r2 SA = 4(1/2d)2

  11. Let’s try more: Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing 2 hours later? Answer: 65 mi/h

  12. Let’s try more: A ladder 10 ft. long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft. from the wall? Answer: -3/4 ft/s

  13. A trough is 10 ft long and its ends are in the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 feet. If the trough is filled with water at a rate of 12 feet cubed per minute, how fast is the water level rising when the water is half a foot deep? Answer: 4/5 ft/min

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