4.6: Related Rates. 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.

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Presentation transcript:

4.6: Related Rates

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.

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.

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. )

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.

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

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

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?

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. 

A police car approaches a 4-way intersection from the north chasing a speeding car that has turned the corner and is moving east. When the cruiser is 0.8 mi north of the intersection and the car is 0.6 mi to the east, the police determine using radar that the distance between the two cars is increasing at 15 mph. If the cruiser is going 60 mph what is the speed of the car? Example

Related Rate Example Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

Related Rate Example Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

Related Rate Example A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec. 1.Show how the motion of the two ends of the ladder can be represented by parametric equations. 2.What values of t make sense in this problem. 3.Graph using a graphing calculator. State an appropriate viewing window. 4.Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t =.5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground?

Related Rate Example A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec. 1.Show how the motion of the two ends of the ladder can be represented by parametric equations. 2. What values of t make sense in this problem.

Related Rate Example 3.Graph using a graphing calculator. State an appropriate viewing window. 4. Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t =.5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground? t: [0,4] x: [0,13] y: [0, 13]