2First, 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
3Now, suppose that the radius is changing at an instantaneous rate of 0 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 ofNote: This is an exact answer, not an approximation likewe got with the differential problems.
4Water is draining from a cylindrical tank at 3 liters/second Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?Find(We need a formula to relate V and h. )(r is a constant.)
5Steps 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.
6Ex. 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.Givens:Given equation:Differentiate:
7Hot Air Balloon Problem: Given:How fast is the balloon rising?Find
8Hot Air Balloon Problem: Given:How fast is the balloon rising?Find
9The velocity of an airplane tracked by radar An airplane is flying at an elevation of 6 miles on a flightpath that will take it directly over a radar tracking station.Let s represent the distance (in miles)between the radarstation and the plane. If s is decreasing at a rate of 400miles per hour when s is 10 miles, what is the velocity ofthe plane.s6x
10Given:Find:Equation:Solve:To find dx/dt, wemust first find xwhen s = 10
11A fish is reeled in at a rate of 1 foot per second from a bridge 15 ft. above the water. At whatrate is the angle between the line and the waterchanging when there is 25 ft. of line out?x15 ft.
13Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. 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?BA
14Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. 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?BA