Related Rates and Applications Lesson 3.7

Related Rates Guidelines Draw a figure, if appropriate, and assign variables to the quantities that vary. (Be careful not to label a quantity with a number unless it never changes in the problem) Find a formula or equation that relates the variables. (Eliminate unnecessary variables) Differentiate the equations. (typically implicitly) Substitute specific numerical values and solve algebraically for any required rate. (The only unknown value should be the one that needs to be solved for.)

General vs. Specific Note the contrast … General situation properties true at every instant of time Specific situation properties true only at a particular instant of time We will consider a rock dropped into a pond … generating an expanding ripple

Example 1 At the point in time when r = 8 radius is increasing at 3 in/sec That is we are given We seek the rate that the area is changing at that specific time We want to know r = 8

Solution Strategy Draw a figure label with variables do NOT assign exact values unless they never change in the problem Find formulas that relate the variables A r

Solution Strategy Differentiate the equation with respect to time Substitute in the given information

Example 2 Given Find when x = 3 Note: we must differentiate implicitly with respect to t

Example Now substitute in the things we know Find other values we need when x = 3, 32 + y2 = 25 and y = 4

Example Result −2∗3∗4 2∗4 =−3

Example 3 A bag is tied to the top of a 5 m ladder resting against a vertical wall. Suppose the ladder begins sliding down the wall in such a way that the foot of the ladder is moving away from the wall. How fast is the bag descending at the instant the foot of the ladder is 4 m from the wall and the foot is moving away at the rate of 2 m/s? Find the rates by differentiating both sides. 5 m y Ladder Chain Rule Substitute the known information x Using The Pythagorean Theorem, the equation is: Find other important values: Solve for the unknown m/s x

Example 4 A spherical balloon is being filled with a gas in such a way that when the radius is 2ft, the radius is increasing at the rate 1/6 ft/min. How fast is the volume ( ) changing at this time? Find the derivative by differentiating both sides. Chain Rule Substitute the known information ft3 per minute Solve for the unknown

Example 5 A rocket launches with a velocity of 550 miles per hour. 25 miles away there is a photographer filming the launch. At what rate is the angle of elevation of the camera changing when the rocket achieves an altitude of 25 miles? Find the rates by differentiating both sides. This is “x” and there is no “x” in the derivative… x Chain Rule Substitute the known information Θ 25 mi Using The Trigonometry, the equation is: Use “x” to find other important values: Solve for the unknown rad/h

Example 1 A person 6 ft tall is walking away from a streetlight 20 ft high at the rate of 7 ft/s. At what rate is the length of the person’s shadow increasing? Find the rates by differentiating both sides. 20 ft 6 ft Chain Rule y x Substitute the known information Using similar triangles, the equation is: Solve for the unknown ft/s

Example 3 A trough 10 ft long has a cross section that is an isosceles triangle 3 ft deep and 8 ft across. If water flows in at the rate 2 ft3/min, how fast is the surface rising when the water is 2 ft deep? Find the rates by differentiating both sides. 10 ft 8 ft Nothing is known about b… b 3 ft h Using similar triangles: Chain Rule Substitute the known information Using the volume of a prism, the equation is: Solve for the unknown ft/min

Truck 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? B A

p 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 p

Electricity R1 The combined electrical resistance R of R1 and R2 connected in parallel is given by R1 and R2 are increasing at rates of 1 and 1.5 ohms per second respectively. At what rate is R changing when R1 = 50 and R2 = 75? R2

Draining Water Tank Radius = 20, Height = 40 The flow rate = 80 gallons/min What is the rate of change of the radius when the height = 12?

Draining Water Tank At this point in time the height is fixed Differentiate implicitly with respect to t, Substitute in known values Solve for dr/dt

Assignment Lesson 3.7 Page 187 Exercises 1 – 7 odd, 13 – 27 odd