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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 1. Draw a diagram. 2. Represent the given information and the unknowns by mathematical symbols. 3. 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.) 4. Differentiate each term with respect to time (t). (you will use a form of implicit differentiation) 5. Substitute all known values and know rates of change into the resulting equation. 6. Solve the resulting equation for the desired rate of change. 7. Write the answer with units of measure. 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 y x A Given: dz 2 in sec dt z z 72 in. B dx 3 in sec dt x 96 in. dy Find dt x 2 z 2 y 2 Differentiate with respect to time. dx dz dy 2x 2z 2y Divide out a “2”. dt dt dt 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 y x A z x2 z 2 y2 (96) (72) y 2 120 y Given: dz 2 in sec dt 2 z 72 in. B dx 3 in sec dt x 96 in. dx dz dy x z y dt dt dt dy 96(3) 72(2) y dt dy Find dt Plug in known values. What is y? 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 y x A Given: dz 2 in sec dt z z 72 in. B dx 3 in sec dt x 96 in. dy 96(3) 72(2) 120 dt 1.2 in dy sec dt dy Find dt Solve for dy/dt 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. 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? 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?