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4.6 Related Rates, p. 246

AP Calculus AB/BC. 4.6 Related Rates, p. 246. Consider a sphere of radius 10cm. The volume would change by approximately. First, a review problem:. If the radius changes 0.1cm (a very small amount) how much does the volume change?. The sphere is growing at a rate of.

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4.6 Related Rates, p. 246

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  1. AP Calculus AB/BC 4.6 Related Rates, p. 246

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

  3. The sphere is growing at a rate of . Note: This is an exact answer, not an approximation like we got with the differential problems. 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.)

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

  5. 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.

  6. 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

  7. 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

  8. 45 m 6 m Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?

  9. Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?

  10. Draining Conical Reservoir Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment?

  11. Example 2: Hot Air Balloon Problem Given: How fast is the balloon rising? Find

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

  13. Sliding Ladder. A 13-ft ladder is leaning up against a house when its base starts to slide away. By the time the base is 12-ft from the house, the base is moving at the rate of 5 ft/sec. • How fast is the ladder sliding down the wall at that moment? • At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment? • At what rate is the angle θ between the ladder and the ground changing at that moment?

  14. (a) How fast is the ladder sliding down the wall at that moment? Substitute the known values.

  15. (b) At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment?

  16. (c) At what rate is the angle θ between the ladder and the ground changing at that moment? p

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