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4.6: Related Rates Olympic National Park, Washington Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington 4.6: Related Rates Olympic National Park, Washington Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington 4.6: Related Rates Olympic National Park, Washington Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington 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? 4 V r3 3 dV 4 r 2 dr dV 4 10cm 0.1cm 2 dV 40 cm3 The volume would change by approximately 40 cm 3 . 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 V r3 3 dV dr 4 r 2 dt dt dV cm 2 4 10cm 0.1 dt sec dV cm3 40 dt sec The sphere is growing at a rate of 40 cm3 / sec . 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? 3 cm dV L 3000 3 sec dt sec dh Find dt (We need a formula to relate V and h. ) V r 2h dV 2 dh r dt dt cm3 2 dh 3000 r sec dt (r is a constant.) cm3 3000 dh sec dt r2 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: d 4 rad 0.14 dt min How fast is the balloon rising? dh Find dt h tan 500 d 1 dh 2 sec dt 500 dt 1 dh sec 0.14 4 500 dt h 500ft 2 Hot Air Balloon Problem: Given: d 4 rad 0.14 dt min 2 How fast is the balloon rising? dh Find dt h tan 500 d 1 dh 2 sec dt 500 dt 1 dh sec 0.14 4 500 dt 2 1 4 1 sec 4 h 500ft 2 dh 2 0.14 500 dt 2 ft dh 140 min dt 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? r t d 1 40 4 10 1 30 3 10 32 42 z 2 B z 5 y3 A x4 9 16 z 2 25 z 2 5 z 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? r t x d y z 2 2 2 1 1 dz dx dy 40 2 x 10 42 y 30 10 2z 3 dt dt dt 32 42 z 2 dz 4 40 3 30 2 5 9 16 z dt 2 dz 25 z 250 5 dz dt 50 5 z dt B z 5 y3 dy 30 dt A x 4 dx 40 dt miles 50 hour