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2.6 Related Rates

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First, a review problem: Consider a sphere of radius 10cm. If the radius changes 0.1cm (a very small amount) how much does the volume change? The volume would change by approximately.

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Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec. (Possible if the sphere is a soap bubble or a balloon.) The sphere is growing at a rate of. Note: This is an exact answer, not an approximation like we got with the differential problems.

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Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping? Find ( r is a constant.) (We need a formula to relate V and h. )

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Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

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Ex. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When this radius is 4 ft., what rate is the total area A of the disturbed water increasing. Given equation: Givens: Differentiate:

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Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

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Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

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The velocity of an airplane tracked by radar An airplane is flying at an elevation of 6 miles on a flight path that will take it directly over a radar tracking station. Let s represent the distance (in miles)between the radar station and the plane. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the velocity of the plane. s x 6

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Given: Find: Equation: Solve: To find dx/dt, we must first find x when s = 10

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A fish is reeled in at a rate of 1 foot per second from a bridge 15 ft. above the water. At what rate is the angle between the line and the water changing when there is 25 ft. of line out? 15 ft. x

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Given: Find: Equation: Solve: x = 25 ft. h = 15 ft.

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B A Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later?

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B A Truck Problem: How fast is the distance between the trucks changing 6 minutes later? Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.

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