Math 103: Related Rates
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1 Math 103: Related Rates Ryan Blair University of Pennsylvania Thursday October 20, 2011 Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
2 Outline 1 Review 2 Related Rates Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
3 Review Important Formulas from Last Time 1 d dx (f 1 1 (x)) = f (f 1 (x)) 2 d dx (ln(x)) = 1 x d 3 dx (sin 1 (x)) = 1 1 x 2 4 d dx (tan 1 (x)) = 1 1+x 2 5 d dx (sec 1 (x)) = 1 x x 2 1 Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
4 Related Rates is the most important application of calculus we have seen so far. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
5 Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 10 cm3 s. How fast is the radius of the balloon increasing when the diameter is 4cm? Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
6 Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 10 cm3 s. How fast is the radius of the balloon increasing when the diameter is 4cm? How To Approach These Problems 1 Draw a picture and name the variables and constants. 2 Write down any additional numerical info. 3 Write down what you are asked to find. 4 Write an equation that relates the quantities. 5 Differentiate with respect to t. 6 Finish solving the problem. Remember units. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
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. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
8 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 Draw a picture and name the variables and constants. 2 Write down any additional numerical info. 3 Write down what you are asked to find. 4 Write an equation that relates the quantities. 5 Differentiate with respect to t. 6 Finish solving the problem. Remember units. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
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? Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
10 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 Draw a picture and name the variables and constants. 2 Write down any additional numerical info. 3 Write down what you are asked to find. 4 Write an equation that relates the quantities. 5 Differentiate with respect to t. 6 Finish solving the problem. Remember units. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
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. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
12 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 Draw a picture and name the variables and constants. 2 Write down any additional numerical info. 3 Write down what you are asked to find. 4 Write an equation that relates the quantities. 5 Differentiate with respect to t. 6 Finish solving the problem. Remember units. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
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. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
14 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 Draw a picture and name the variables and constants. 2 Write down any additional numerical info. 3 Write down what you are asked to find. 4 Write an equation that relates the quantities. 5 Differentiate with respect to t. 6 Finish solving the problem. Remember units. Ryan Blair (U Penn) Math 103: Related Rates Thursday October 20, / 9
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