R ELATED R ATES

The Hoover Dam

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? Example of a Related Rate:

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 1: Read the problem carefully. Step 4: Write an equation relating the quantities. Step 5: Implicitly differentiate both sides of the equation with respect to time, t.

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

The table below lists examples of mathematical models involving rates of change. Let’s translate them into variable expressions: Verbal Statement:Mathematical Model Water is being pumped into a swimming pool at a rate of 10 cubic meters per hour. The velocity of a car is 50 miles per hour The length of a rectangle is decreasing at a rate of 2 cm/sec.

C = 2  r A =  r 2 V = 4 / 3  r 3 SA = 4  r 2 a 2 + b 2 = c 2 a b c r r h V =  r 2 h r h V = 1 / 3  r 2 h A = 1 / 2 bh h b 30  60  x x/2x/2 x / 2  √3 r

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? Let’s try: What are we trying to find? What variable can we assign this unknown? dA dt =? What formula can I use? A =  r 2 How can I get dA / dt out of that formula? dA / dt = 2  r dr / dt Substitute in what you know! dA / dt = 2  (30 m)(1 m/s) dA / dt = 60  m 2 /s

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 / 2 m/s, how fast is the area of the ripple increasing when the radius of the ripple is 20 m? Answer: 20  m 2 /s or 62.8 m 2 /s

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

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

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

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