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4.6 Related Rates

4.6 Related Rates. What are related rates problems?. If several variables that are functions of time t are related by an equation, we can obtain a relation involving their time rates by differentiating with respect to t. General Process for Solving Related Rate Problems. Draw a diagram.

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4.6 Related Rates

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  1. 4.6 Related Rates

  2. What are related rates problems? • If several variables that are functions of time t are related by an equation, we can obtain a relation involving their time rates by differentiating with respect to t.

  3. General Process for Solving Related Rate Problems • Draw a diagram. • Represent the given information and the unknowns by mathematical symbols. • Write an equation involving the rate of change to be determined. (If the equation contains more than one variable, it may be necessary to reduce the equation to one variable.) • Differentiate each term with respect to time (t). (you will use a form of implicit differentiation) • Substitute all known values and know rates of change into the resulting equation. • Solve the resulting equation for the desired rate of change. • Write the answer with units of measure.

  4. Example • If one leg AB of a right triangle increases at the rate of 2 inches per second, while the other leg AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 72 inches and AC = 96 inches. C Given: y x Find B A z Differentiate with respect to time. Divide out a “2”.

  5. Example • If one leg AB of a right triangle increases at the rate of 2 inches per second, while the other leg AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 72 inches and AC = 96 inches. C Given: y x Find Plug in known values. B A z What is y?

  6. Example • If one leg AB of a right triangle increases at the rate of 2 inches per second, while the other leg AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 72 inches and AC = 96 inches. C Given: y x Find Solve for dy/dt B A z

  7. Example • The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking out at the rate of ½ cubic inch per second. Find the rate at which the water level is dropping when the diameter of the surface is 2 inches.

  8. Example • A bouillon cube with side length 0.8 cm is placed into boiling water. Assuming it roughly resembles a cube as it dissolves, at approximately what rate is its volume changing when its side length is 0.25 cm and is decreasing at a rate of 0.12 cm/sec?

  9. Example • A 20-foot extension ladder propped up against the side of a house is not properly secured, causing the bottom of the ladder to slide away from the house at a constant rate of 2 ft/sec. How quickly is the top of the ladder falling at the exact moment the base of the ladder is 12 feet away from the house?

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