Math 103 Day 9: Related Rates
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1 Math 103 Day 9: Related Rates Ryan Blair University of Pennsylvania Tuesday October 7, 2010 Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
2 Outline 1 Related Rates Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
3 Related Rates is the most important application of calculus we have seen so far. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
4 Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 10 cm s. How fast is the radius of the balloon increasing when the diameter is 4cm? Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
5 Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 10 cm s. How fast is the radius of the balloon increasing when the diameter is 4cm? How To Approach These Problems 1 What quantities are given in the problem? 2 What is the unknown? 3 Draw a picture of the situation with labels. 4 Write an equation that relates the quantities. 5 Finish solving the problem. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
6 Example A water tank has the shape of an inverted circular cone with base radius 2 meters and a height of 3 meters. If the water is being pumped into the tank at a rate of 3 m3 min, find the rate at which the water level is rising when the water is 2 meters deep. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
7 Example A water tank has the shape of an inverted circular cone with base radius 2 meters and a height of 3 meters. If the water is being pumped into the tank at a rate of 3 m3 min, find the rate at which the water level is rising when the water is 2 meters deep. 1 What quantities are given in the problem? 2 What is the unknown? 3 Draw a picture of the situation with labels. 4 Write an equation that relates the quantities. 5 Finish solving the problem. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
8 ExampleA ladder 6ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of.5 ft sec, how fast is the top of the ladder sliding when it is 1ft above the ground? Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
9 ExampleA ladder 6ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of.5 ft sec, how fast is the top of the ladder sliding when it is 1ft above the ground? 1 What quantities are given in the problem? 2 What is the unknown? 3 Draw a picture of the situation with labels. 4 Write an equation that relates the quantities. 5 Finish solving the problem. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
10 Example A round oil slick uniformly 0.1cm thick is being fed by a leak in an off shore oil rig at a rate of 2 m3 sec. Sea turtles have bad eyesight and only see the oil as it is nearly on top of them. If sea turtles swim at a rate of 1 m sec and begins swimming away from the slick as they see it approaching, how far away from the oil rig does a turtle need to be to avoid being overcome by the slick. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
11 Example A round oil slick uniformly 0.1cm thick is being fed by a leak in an off shore oil rig at a rate of 2 m3 sec. Sea turtles have bad eyesight and only see the oil as it is nearly on top of them. If sea turtles swim at a rate of 1 m sec and begins swimming away from the slick as they see it approaching, how far away from the oil rig does a turtle need to be to avoid being overcome by the slick. 1 What quantities are given in the problem? 2 What is the unknown? 3 Draw a picture of the situation with labels. 4 Write an equation that relates the quantities. 5 Finish solving the problem. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
12 Example A light house is located on small island 3km away from the nearest point P on a straight shore line and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1km from P. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
13 Example A light house is located on small island 3km away from the nearest point P on a straight shore line and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1km from P. 1 What quantities are given in the problem? 2 What is the unknown? 3 Draw a picture of the situation with labels. 4 Write an equation that relates the quantities. 5 Finish solving the problem. Ryan Blair (U Penn) Math 103 Day 9: Related Rates Tuesday October 7, / 8
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