Related Rates 1 The Basic Question If one quantity is changing in an equation or situation at a particular moment, how much is another quantity changing The Problem We are given one rate, a relationship relating variables, and asked to find another rate 2 Remarks We usually have three or more variables For example, we might have two dependent variables (for example x and y), and one independent variable (for example t) We are given the relation between two variables (for example x and y) We are given one rate (for example dx/dt) at specific values (for example x, y, and z values) We are asked to find the other rate (ex: dy/dt) 3 1 With Related Rates We use Liebniz’s notation of dy/dt We take derivatives implicitly We know standard formulas for perimeter, surface area, area, and volume for common geometric figures We know the Pythagorean Theorem We draw sketches to understand and solve We must use the correct units 4 Related Rates Process 1. Draw a picture (essential) include all variables 2. Do not label variables as constants 3. Translate to a math problem – search for equations that relate quantities 4. Implicitly differentiate the equation 5 Remarks The key to solving related rates problems is to find an equation which relates the quantities whose rates of change are known to the quantity whose rates of change we would like to find. Then we differentiate this equation implicitly with respect to the appropriate variable (typically t, for time). Finally, we plug in all of our known values so that we can solve for the rate of change we want. 6 2 Example Suppose that the radius r and the circumference of a circle are both functions of time Write an equation that relates dC/dt to dr/dt C 2 r dC dr 2 dt dt 7 Example Suppose that the radius r and the volume of a sphere are both functions of time Write an equation that relates dV/dt to dr/dt V 4 r 3 3 dV dr 4 r 2 dt dt 8 Example Each side of a square is increasing at a rate of 2 cm/sec How fast is the perimeter changing? P 4x dP dx 4 dt dt x dP cm 42 8 dt s 9 3 Example A snowball melts at 2 ft3/h. Assume it is spherical. Find the rate the radius is changing when the radius is 20 inches. 10 Snowball melting Example r r 4 V r3 3 20 inches 1 ft 5 ft 1 12 inches 3 Snowball melting dV 2 dt 11 Snowball melting Example 4 V r3 3 Implicit differentiate with respect to time dV dr 4 r 2 dt dt 2 5 dr 2 4 3 dt dr 9 ft dt 50 hour See http://www.youtube.com/watch?v=zXrm-Smy0qU for another example 12 4 Example A 5 ft tall Calculus student walks away from a lamp post at 7 ft/s. The lamp post is 20 ft tall. When she is 8 ft from the lamp post, find the rate at which the tip of her shadow is moving. 13 Example Calculus student Shadow Tip 14 Example Calculus student Sketch zx 20 5 x z 15 5 Example Calculus student Compare similar triangles 20 5 zx z and develop equation z zx 20 5 3z 4 x 16 Example Calculus student Differentiate 3 y 4 x 3 dy dx 4 dt dt Solve for dy dt dz 4 28 ft 7 dt 3 3 sec 17 Example A 5 ft tall Calculus student walks away from a lamp post at 7 ft/s. The lamp post is 20 ft tall. When she is 8 ft from the lamp post, find the rate at which the length of her shadow is changing. 18 6 Example Calculus student Shadow Length 19 Example Calculus student Sketch The length of the shadow is represented by y 20 y 5 x x y 20 Example Calculus student Compare similar triangles 20 5 x y y and develop equation x y y 3y x 20 5 21 7 Example Calculus student Differentiate 3 y x 3 dy dx dt dt dy dt Solve for dz 1 7 ft 7 dt 3 3 sec 22 Example Calculus student x y z Compare Problems 20 dx dy dz dt dt dt y 5 7 7 28 3 3 x z See http://www.youtube.com/watch?v=JKXgOIJ_N9E for another example 23 Example A car traveling west on I-10 passes a highway patrolman parked 90 feet north of the interstate When a car is 150 feet from the patrolman’s car, radar indicates the car is approaching at 72 ft/sec How fast is the car traveling? 24 8 Example police car Sketch x 90 y 25 Example police car Find the equation and differentiate x 90 y 2 902 x2 y 2y y dy dx 2x dt dt dy dx x dt dt 26 Example x police car 90 y y 2 902 1502 y 120 dy dx x dt dt dy 120 150 72 dt dy feet 90 dt sec (about 61 mph) y 27 9 Example police car For more information try https://www.youtube.com/playlist?list=PL5KkMZvBpo 5Ahgc5EnzYPYjBqz3utcPNv This link is to a video series of related rates examples, including the police car problem 28 Example A 26 foot ladder leans against a vertical wall The ladder is sliding down When the foot of the ladder is 10 ft from the base of the wall it is moving at 4 ft/sec How fast is the top of the ladder moving down the wall at that instant? 29 Example slipping ladder x 2 y 2 262 Sketch 26 y x dx dy 2y 0 dt dt dx dy 2 x 2 y dt dt 2x x 10 y 262 102 24 dx 4 dt 30 10 Example x dx dy y dt dt slipping ladder x 10, y 24 dx 4 dt 4 10 24 dy dt dy 5 ft dt 3 sec See http://www.youtube.com/watch?v=m7qZDl_82Wo for another example 31 Example The cost C (in dollars) of manufacturing x number of high-quality computer laser printers is C(x) = 15x4/3 + 54x2/3 + 600,000 Currently, the level of production is 1728 printers and that level is increasing at the rate of 350 printers each month. Find the rate at which the cost is increasing each month 32 Example laser printers C x 15 x 4 3 54 x 2 3 600000 dx dC =350 when x 1728 and we are to find dt dt dC dx 20 x1 3 36 x 1 3 dt dt dC 13 1 3 20 1728 36 1728 350 dt $85,050 per month 33 11 Example The monthly revenue R (in dollars) of a telephone polling service is related to the number x of completed responses by the function R x 25 3.5x 2 25x 12,000 on 0,1500 If the number of completed responses is increasing at the rate of 10 forms per month, find the rate at which the monthly revenue is changing when x = 750 Stop polling me! 34 Example telephone polling R x 25 3.5 x 25 x 12, 000 on 0,1500 2 dx forms 10 when x 750 dt month R 25 3.5 x 2 25 x 12,000 12 1 2 dR 25 dx 3.5 x 2 25 x 7 x 25 dt 2 dt 1 2 25 2 3.5 750 25 750 7 750 2510 2 $469.73 per month Example 35 A pile of sand is shaped like a right circular cone. A pile of sand is shaped like a right circular cone. It is shaped such that the base radius of the pile is one third of its height. Sand is being added to the pile at the steady rate of 10 ft3/min. How fast is the height of the pile rising when the pile is 4 feet tall? 36 12 Example A pile of sand is shaped like a right circular cone. We are given 1 V r 2h 3 1 r h 3 dV ft 3 10 dt min dh when h 4 ft dt We are asked to find 1 1 3 V h h h 3 3 27 2 Writing V in terms of h Taking the derivative implicitly 37 A pile of sand is shaped like a right circular cone. Example Solving for dV 2 dh h dt 9 dt dV 2 dh h dt 9 dt dh dt 10 9 4 2 dh dt 45 ft dh 90 8 min dt 16 38 Example A bird is flying horizontally 40 feet above your head at 20 ft/s How fast is the angle of elevation changing when your horizontal distance from the bird is 30 feet? 40 ft 39 13 bird flying Example Sketch x 2 40 40 x x 30 x2 40 50 40 bird flying Example dx ft 20 dt sec 40 From sketch tan x Given x 30 ft, 50 40 Differentiate 30 From sketch sec sec2 d 40 dx 2 dt x dt 5 25 sec2 3 9 41 Example Substitute know values into sec2 bird flying d 40 dx 2 dt x dt 40 5 d 20 2 2 dt 30 2 d 8 radians dt 25 second 42 14 Two Ships A ship is 12 miles due north of port and is traveling north (away from port) at a rate of 4 miles per hour (mph). Another ship is 5 miles due east of port and is traveling west (back towards the port) at a rate of 3 mph. What is the rate of change of the distance between the ships? 43 Ship A c 122 52 13 c = 13 We want to find a=12 Port b=5 dc dt Ship B We note that Ship A is moving away from port faster than Ship B is moving towards it, so we believe the distance between the two ships is increasing. 44 We are given that Ship A is moving away from port at a rate of 4 mph. This means that the rate of change of distance a between Ship A and the port is increasing, so da 4 dt We know that Ship B is moving towards port at a rate of 3 mph. This means that the rate of change of distance (b) between Ship B and the port is decreasing, so db 3 dt 45 15 Finally, we need an equation that relates a, b, and c. The most obvious one that comes to mind is the Pythagorean Theorem which relates the lengths of the sides of a right triangle: c 2 a 2 b2 Using implicit differentiation with respect to time, t, we get: 2c dc da db 2a 2b dt dt dt 46 2c dc da db 2a 2b dt dt dt Since we want to solve for we can divide both sides by and then plug in all known values. da db b dt dt c dc 2(12)(4) 2(5)(3) 2.54 dt 2(13) dc a da b db dt c dt c dt a This tells us that the distance between the two ships is increasing at a rate of 2.54 mph 47 What if Ship A was traveling due south (back towards port) at a rate of 4 mph? In this case, all of the values are da 4 exactly the same except dt Then dc 2(12)(4) 2(5)(3) 4.85 dt 2(13) This time the distance between the two ships is decreasing at a rate of 4.85 mph. Since both ships are heading back towards port, the distance between them is getting smaller, and at a quicker rate. 48 16 Pythagorean Theorem Problems There are some special triangles that have sides that are integers. If you recognize one these in your problem, it can save work in your solution. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 9, 40, 41 12, 35, 37 You don't need to remember these because you can use the Pythagorean Theorem to solve the triangle. Also multiples of these triangles are often used. 49 Frequently Used Formulas Area of a triangle: Area of a circle: Area of a square: Volume of a sphere: Surface area of a sphere: Surface area of a cylinder Volume of a cylinder: Volume of a cone: Volume of a cube: Surface Area of a Cube: (1/2)bh r2 s2 (4/3) r3 4 r2 2 r h + 2 r2 (both ends) 2 r h (no ends) 2 r h + r2 (one end) r2h (1/3) r2h s3 6 s2 50 Remarks Related Rates Tutorials and Animations http://astro.ocis.temple.edu/~dhill001/relate drates/relatedrates.html http://www2.sccfl.edu/lvosbury/CalculusI_Folder/RelatedRate Problems.htm http://people.hofstra.edu/Stefan_Waner/Real World/tutorials/frames4_4.html http://www.karlscalculus.org/calc8_1.html 51 17

1/--pages