Related Rates

1. Related Rates Alissa Johnson and Tommy Martin

2. Table of Contents Title Slide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Related Rates: A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .3 Real World Applicability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Mathematician Spotlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Basic Analytical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 AP Multiple Choice Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conceptual Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Graphical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Free Response Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Works Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

3. Related Rates: A Brief Introduction For instance: The first step in most related rates problems is finding an equation that involves all of the variables which will be used in the problem. This could be an area or volume formula, as it is in some cases. It could also involve triangle trigonometry or some other given equation. Next, implicitly differentiate the equation with respect to time. Don’t forget basic derivative rules (chain rule, product rule, quotient rule, etc.)(Sometimes some algebraic manipulation is necessary or helpful prior to this step). Finally, use the given values from the problem in order to solve for the requested quantity or rate!

4. Real World Applicability Related rates have infinite applications. In fact related rates are often referred to as the application of the derivative, which means that a related rates problem by definition is a real-world application! How exciting! As stated before, anything that can be mapped by an equation can be made into a related rates problem. In other words, anything changing with respect to time can be related to something else changing with respect to time. These relations are important across all disciplines, and most often used in physics. They could be crucial for things such as the launch of a rocket. Indeed, the rocket equation which relates the rate at which a rocket is losing mass to its acceleration, is derived using related rates as are so many other equations describing fundamental concepts in physics. The concept of related rates makes is easier to understand and use equations for displacement, velocity, and acceleration. Related rates applications are truly limitless. Who does not want to know how quickly the area of a halfpenny expands when placed on a hot shovel? William Ritchie was certainly curious (more about him later). Together, with the aid of related rates, we can solve at what rate the water in a pool is rising, the rate at which Erica moves when she runs in a circle, or the rates at which a rebel starship and debris fly away from an exploding Death Star!

5. Mathematician Spotlight The man who brought related rates to their glory goes (or went if you do not believe the fathers of Calculus are immortal) by the name of Rev. William Ritchie. He actually started out in the church, hence the title Reverend. After attending some conferences in Paris, he found out that he had a knack for mathematics and eventually became a professor of natural philosophy at London University. He published a paper, Principles of Differentiable and Integral Calculus, which included many examples focusing on related rates. Ritchie’s goal was to make Calculus less abstract. He wanted to make it understandable and applicable to people with a simple math background. Which was an excellent idea, because related rates are so applicable to everyday life!

6. Analytical Example The volume of a cylinder is defined by the following… Some house elves are making a BIG cake for the end of the year celebration. When the cake is baking it expands. However, because of the container it is in, it only expands upward. Given that the radius starts at 25 feet, at what rate will the volume be increasing when the height is increasing at a rate of 5 ft/s.

7. Working through the example Start with the given equation Take the derivative of the equation with respect to time (t). This will be done using the product rule. We know that the radius is not increasing so therefore as the over any change in time (dt) the change of the radius (dr) will be zero.

8. Continued We are given all the remaining problems in the problem. So just plug everything in and arrive at the answer! That’s one fast growing cake!

9. Multiple Choice Example Sandy is preparing her rocket for a launch to the moon. She is taking it on a test drive around Bikini Bottom (which is obviously completely plausible). Plankton tries to thwart her plan by using a giant fan to create a current in the water and drive the rocket off course. The rocket is launched from position (0,0) on Sandy’s Map with a constant upward velocity of 20 m/s. Plankton’s fan creates a constant current in the water directed due east at 500 m/s. When ѳ = 45 Sandy notices the rocket is at position (10, 10). What is the rate that the angle, ѳ, between the rocket and the ground is changing when ѳ = 45°? x A) 24 °/s B) °/s C) 26 °/s D) -24 °/s E) -26 °/s y ѳ

10. Solution Process First, we need to establish an equation that relates all the variables we wish to use and find. Then, take the derivative of that equation with respect to time. Don’t forget quotient rule! Finally, solve for dθ/dt and insert known values for the variables in the equation. The final result should be -24°/s, or choice D.

11. Why Are The Other Answers Incorrect? Almost right away, we can eliminate the first 3 answers. Conceptually, we can tell that the rocket is being pushed MUCH faster in the x direction than it is propelling itself in the y direction. Since the value of x is increasing faster than the value of y (), the angle is decreasing! We can thus conclude that the rate that the angle is changing is negative. Analytically, we can also see that, when we take the derivative, we know that is far greater than (and the x and y values are positive) and thus is a negative value. Both D and E solutions are plausible answers! The true answer is only revealed through correct calculations. *If you chose choice E, you probably used the quotient rule incorrectly* *If you chose choice B, you likely differentiated tan incorrectly and/or used cosinstead of sec!* *If you arrived at choice A your solution process was correct, but check sign conventions!* *If you arrived at choice C, be wary of quotient rule AND sign conventions!*

12. Conceptual Example Captain Jack Sparrow was admiring his reflection for a bit too long, and did not notice his ship scraping across a rocky cliff. Unfortunately, he notices too late. There is a hole and his ship has already begun to fill with water. Conveniently, the bottom of the ship is rectangular. Jack Sparrow is unaware of the exact dimensions of his ship; however, he does know that the length is equal to three times the width and the height is twice the width. Knowing this, he wonders what the rate of change of the total volume of the water in the ship would be at the instant when the height of the water is equal to the width of the ship. (*Note* Only the height of the water entering the ship is changing.)

13. Solution Process First, we need to establish an equation that relates all the variables. In this case, volume because we are looking for the volume of the water in the ship. Then, use the given information to consolidate into two variables. Be careful not to be thrown off and put everything in terms of width (this will create a problem later since dw/dt is zero, yet the height is changing). Next, differentiate the new equation with respect to time. Don’t forget product rule! Finally, we can use the rest of the given information to further simplify the equation. Using the fact the width of the water is not changing eliminates the second term. Also, we want the rate of change of the volume when the height is equal to the width. So, use that information to get a final equation for Dv/dt in terms of height and the rate of change of the height.

14. Graphical Example R2D2 and C3PO are taking the Millennium Falcon out for a routine mission. They are paying careful attention to the radar screen and see the radar arm constantly sweeping out area of the polar curve . Additionally, due to their attentiveness they also happen to notice a fleck of dust moving across the screen. Coincidentally, it appears to be moving according to the function , so that at time t seconds . They then wonder, is there a time,t, that the rate at which the radar arm is sweeping out area is equal to the horizontal velocity of the dust particle? (If so, when is the first time this occurs?) (Honestly, who doesn’t?)

15. Dust particle

16. SOLUTION! First, we must find an equation to represent the area being swept out by the radar arm. Since it is a polar curve the formula to be used is Next, we need to get the area with respect to time. Use the given information and differentiate to get an expression for d in terms of dt. Then, use the previous steps to plug in and obtain an equation for A(t). Next, we need an equation for x with respect to time. It is easier to start with x(). Again, using the given information we can then get an equation for x(t).

17. Continued . . . Now, we are trying to relate the rates so we must differentiate both expressions with respect to time. For A(t), don’t forget the chain rule and the FTC! Also beware of multiple chain rules when differentiating x(t). Next, since we want to know when the horizontal velocity (dx/dt) is equal to the rate of change of area (dA/dt) we set them equal to each other. Finally, use the intersect (zero) feature of your graphing calculator to find the first value of t where the two are equal if such a t exists. The two graphs intersect only at t=0.

19. Free Response Example Batman and JangoFett both are at point (0,0) at time 0. JangoFett flies straight up perpendicular to the ground with a velocity modeled by. Batman runs at 4 miles per hour along the ground. JangoFett’s missile can only launch 1 mile. a) Find the distance both have traveled since the starting point at the when they are 1 mile apart b) Find the rate that the distance between them is increasing at the maximum distance apart within the range that Batman can get hit (when they are 1 mile apart).

20. D J B

21. When solving a related rates problem, the first step will always be finding an equation that relates all of the given values as well as what you are looking for. Then simply take the derivative and solve! For this problem, we have the rates at which both characters are moving. This means that, if we can find the time when they are 1 mile apart, we can find the distance they have traveled using integrals. First we need to find an equation that relates all our values. Good ole Pythagorean At some time, x, the two men will be 1 mile apart. The distances traveled can be represented by the integral of their rates.

22. Solve the integrals Use your graphing calculator’s intersect function to find the answer. Solve for Batman’s distance (plus zero because he was initially at point 0) Solve for Jango’s distance (adding zero again for the same reason)

23. Find the equation that relates all the variables (and their rates) that we wish to use Take the derivative Solve for the rate the distance is changing Plug in all known values. The distances we solved for in part a will go in for variables J and B. The rates are equivalent to their velocities Plug it in to the calculator! Is JangoFett skilled enough to make this shot?

24. Analytical Problems FOR YOU TO TRY! 1) In an intense game of Quidditch, Harry spots the snitch. Realizing he will not be able to get there on time, Draco casts a spell on the snitch. This spell causes the circumference of the spherical snitch to swell in size at a rate of 21 centimeters per second. Harry can only hold a snitch with a volume of 15 cubic centimeters. When the volume is at the maximum that Harry can hold, at what rate is the volume changing? 2) The Sarlacc Pitt quite obviously takes the shape of a cone (point down). The equation for the volume of a cone is . For this particular cone the height is equal to the radius for all points in time. When Luke is escaping from Jabba, he is killing his guards left and right and the pit is filling up! If the height (that has been filled) is 4 meters and the radius is growing at 9 meters per second, at what rate is the volume changing? 3) Hawkeye is standing on top of a roof that is 120 feet tall. Loki is flying his speeder parallel to the ground at a height of 53 feet moving in a straight path away from the building. Hawkeyes has a hawk’s eye and can tell that the distance between him and Loki is changing at 31 feet per second. If Loki’s speeder travels at 45 feet per second, how far does the arrow have to travel when Loki is 100 feet away from the building?

25. 4) The Lego Starwars gang is stuck in the trash compactor! The volume of the trash compactor is quickly decreasing (it takes the form of a rectangular prism). The length of the prism is constant at 25 meters and the height is also constant at 10 meters. To attempt to slow the process of being smashed, they place a metal bar (that conveniently forms an isosceles triangle with is height equal to the height of the prism and its base being equal to the width of the prism). At the time when the area of this triangle is 20 meters squared and the volume is decreasing at a rate of 10 cubic meters per second, what is the rate at which the width is changing? 5) Lego Indiana Jones is running away from a big stone ball down a circular set of stairs with a decreasing radius. The ball can no longer fit down the stairs when the radius is 4 meters. At this time if the radius is decreasing at 7.5 meters per second, at what rate is the circumference decreasing? 6) A Lego rocker is putting everything he’s got into a song. However, he’s not the greatest and as a result fans are leaving the arena. The number of fans in the arena is modeled by the equation , where F is the number of fans, t is the time in minutes, and m is the number of notes missed. If the rocker consistently misses 6 notes per minute, what is the rate that fans are leaving after 10 minutes?

26. 7) The Lego squad is going old school (building at the skill level of a young Tommy Martin). The construction crew decides to build a simple house, just a square base and a flat roof. Every once in a while the crew decides to add another level. Each level is 4 meters tall and the number of layers is modeled by (where t represents time in hours). The base’s dimensions start off as 12 meters by 12 meters, but supplies are running low so the length of both sides of the base is decreasing at a rate of .25 meters per LAYER NOT HOUR. When time is equal to 7 hours and the volume is growing at a rate of 7 , at what rate is the height growing? 8) The Justice Leagues Watchtower has a circular orbits around the earth. It gets closer to and further away from earth depending on the circumstances (international warfare or interplanetary obviously require different distances from earth). The distance from the center of the earth (the orbital’s radius) is represented by Flash is decelerating quickly to avoid crashing into a young child. If Flash is slowing down at and at time zero his velocity is , what is the first time the circumference of the orbital is changing at he same rate that as Flash is moving (his velocity)?

27. 9. Frodo accidentally takes the Ring too close to the fires of Orodruin, and the Ring starts to expand due to this heat exposure. He notices that each radius is increasing at a rate of 0.5mm/s and the height is increasing at a rate of 0.2 mm/s. He wonders at what rate the surface area is increasing at the instant when r₁ = 15mm, r₂ = 13.5mm and the height is 2mm. The surface area of a hollow cylinder is . 10. In an intense Quidditch match against Slytherin, Oliver Wood spots Marcus Flint obtain the Quaffle back near Slytherin’s goal post. When Flint has made it halfway up the pitch, (75m from Gryffindor’s goal post) he is traveling horizontally at a speed of 20 m/s. If the angle his path makes with the top of the tallest goal post (9.1m) is 30⁰. At what rate is the distance between Flint and the top of the post changing? 11. Spongebob’s turbo spatula can crank out crabby patties at a rate of 10 patties/s. The mob of hungry anchovies consumes patties according to the equation patties/s. Where C is the number of customers entering per second. The number of customers entering the KrustyKrab per second is modeled by the piecewise function: . Write An integral expression for the total number of patties at time, t, and use it to find the rate that the total number of patties is increasing at time t=4s? 12. Sandman spots Spidey and the police chasing him and quickly disguises himself among a growing pile of sand nearby (how convenient). The sand is falling, coincidentally, into a square pyramid. The volume of the sand pyramid is increasing at a rate of 400 cm³/s prior to sandman ‘s sand pile appearing directly behind it. If sandman’s sand pile stands 67cm tall, how long before the pyramid in front of him is taller than his sand pile? The initial base of the sand pyramid is 12.5cm and its base is growing at a rate twice that of the height, which is growing at a rate of 0.05cm/s. D ϴ x

28. 13. Yoda is going for a slow stroll and pondering some new Jedi training techniques. A nearby lamp, 365 cm tall, causes him to cast a shadow as he walks. Yoda is leisurely strolling at a rate of 20cm/s , and he stands 66 cm tall. When he is 50 cm from the lamp a) at what rate is the tip of his shadow moving? b) at what rate is the length of his shadow changing? 14. Bored and distracted, Tony Stark decides to take one of his new suits, mark 17, out for a spin. As he is taking off from the ground he is rising at a rate of 30m/s. A local fan/stalker is observing Stark 40 m from his take off point and wants to calculate the rate of change of the angle of elevation at a certain instant. What is the rate of change of the angle of elevation of iron man fro m his stalker when Stark is 300 m above the ground? (in radians/s) 15. On the run, and hunting for horcruxes, Hermoine is in a rush and apparates herself, Ron and Harry to New York City because it just “popped into her head.” When they arrive, the trio encounters Captain America on a night stroll. Startled by the sudden arrival of three people and The loud pop of apparation, he instinctively throws his shield at the three wizards. Ron and Harry turn around and start to run away at a rate of 5m/s as the shield is hurtling towards them at a rate of 10m/s. They are initially 45m from Captain America. Fortunately, Hermoine quick to react and actually uses her want to halt the shield in its tracks before it hits them a) If Hermoine hadn’t stopped the shield, how long after it was thrown would it have hit Harry and Ron b) The shield was also rotating at a rate of 15 rad/s . The radius of the shield is 45 cm. Considering only this rotation, find an expression for the linear rate of change ()of a chip on the edge of the shield with respect to and solve for when = 12rads . 16. Oh no! Bilbo lost his conveniently hobbit sized sword, Sting. Luckily, Gandalf happens to have a spare, but it is too big. He puts a spell on the sword to shrink it down to Bilbo's size. If the surface area of the sword is Given by the equation where A is the area of the top surface of the sword and l and w are the length and width of the blade respectively. The blade width remains a constant 2cm. At what rate is the top surface area shrinking if the length is shrinking at a rate of 4cm/s and the total surface area is shrinking at a rate of 19cm/s? y x s x

29. Multiple Choice Problems 1. Spongebob is on his way to deliver the very first KrustyKrab pizza. Unfortunately, he gets lost along the way. Fortunately though, because Squidward is of no help, he happens across a friendly Gungan General by the name of Jar JarBinks. Even though he is a bit confused himself, Jar Jar agrees to try and help Spongebob deliver the pizza. Somehow, the two are able to make it to the customers house; however, Jar Jar clumsily drops the pizza right before they reach the house. Fortunately though, Spongegbob is the fry cook, and he’s pretty good with people. He explains the situation to the customer and asks to make a fresh new pizza for him, free of charge. When they put the pizza in the oven it has an original radius of 10cm and a thickness of 2cm and the pizza maintains the same proportions throughout baking. If the radius grows at a rate of 0.05cm/s what is the radius of the pie at the instant its volume is changing by 15cm³/s. A. cm B. cm C. 9.156 cm D. 16 cm E. cm

30. 2) Lego Erica is running in a circle away from Two-Face (she figures it will confuse him). When the radius is 17 meters, she will reach her car and be able to get away. If the area of the circle she creates is changing at a rate of 23 meters squared per second, at what rate is the circumference of the circle changing when she reaches her car? A) 18 meters/ second B) -1.353 meters/ second C) 23 meters/ second D) 1.353 meters/ second E) .6765 meters/ second

31. 3) Indiana Jones is trying to escape the clutches of one of his stereotypical enemies. He comes to a ledge (perpendicular to the ground) with a wooden plank leaning against it forming a right triangle with the ground. He leaps on the top of the plank and it begins to slide towards the ground. He deems it safe to jump off this 30 meter long plank only when it is 2 meters off the ground. If the distance between the plank and the wall is increasing at .2 meters per second, what rate is the plank moving down the ledge when he jumps? -2.993 meters/ second 12.007 meters/ second -12.007 meters/ second -3.14 meters/ second -44.8 meters/second

32. 4) The jawa’s are roaming the deserts of Tatooine looking for droids. Today is a bountiful harvest and they are filling up their vehicle quick. The vehicle has a rectangular floor with a width of 17 meters. If the droids are brought in so that the floor fills at 32 meters squared per hour and the width and length grow at a rate of 1.5 meters per hour, what is the length when the vehicle is full in width? A) 38.333 meters B) .837 meters C) 3.213 meters D) 4.333 meters E) - .75 meters

33. 5) Ron is trying to perform the spell WingardiumLeviosabut is not doing well. Hermione is trying to explain that, in order to do the spell, one must flick their wrist so that the change of angle in their wand (relative to a parallel line to the ground) must be precisely Ron’s wand is 10 inches and stays that way throughout the whole spell. If the distance between the tip of the wand and Ron’s hand, distance d, is 1 inch at the end of the spell, at what rate is this distance changing at the end of the spell? d A) 119 inches/second B) -1.2 inches/second C) -11,940 inches/second D) -231 inches/ second E) -199 inches/second

34. Free Response While Golem and Bilbo are fighting over the Ring, they are unaware of the fact that Golem’s cave is slowly filling with water. The bottom of the cave takes on the rough shape of a wide cone. Volume of a cone = • At what rate is the volume of the water in the cave increasing when the height is equal to 120cm if the height of the water is changing at a rate of 0.2cm/s. The dimensions of the cave are r = 3000cm, h=600cm. • At this point the two suddenly notice the water and consequently the ring is launched as a projectile with an initial horizontal velocity of 5m/s and an initial vertical velocity of 3m/s upward. The acceleration due to gravity is 9.8 m/s². Find the position vector of the particle in terms of time (t). • Find the maximum height of the ring and the time where this occurs. Justify your answer. • Find the total distance that the ring traveled from its release to its maximum height.

35. Pictures • marvelsuperheroes.lego.com • allaboutbricks.blogspot.com • www.feralinteractive.com • http://hpandmore.wordpress.com/harry-potter-lego/ • http://nl.lego.wikia.com/wiki/7153_Jango_Fett's_Slave_I • http://www.mynewsdesk.com/be/pressroom/lego-belgique/image/view/lego-the-lord-of-the-rings-frodo-138316 • http://legopiratesthevideogame.wikia.com/wiki/File:Lego-JackSparrow.png • http://www.shopping.com/Lego-SpongeBob-Squarepants-LEGO-SpongeBob-Figure/info • http://lego.wikia.com/wiki/Patrick • http://dragonball.wikia.com/wiki/File:Gotham-City_3-1-.jpg • http://dailyminifig.wordpress.com/category/lego-harry-potter/ • http://www.flickriver.com/photos/24166730@N07/4928549897/ • http://www.shopping.com/Lego-LEGO-SpongeBob-SquarePants-Rocket-Ride/info?sb=1 • http://www.amazon.com/Plankton-LEGO-Spongebob-Squarepants-Piece/dp/B002AIG7AC • http://www.geekalerts.com/lego-pirates-of-the-caribbean-the-black-pearl-4184/ • http://lego.wikia.com/wiki/LEGO • http://shop.lego.com/en-US/Millennium-Falcon-7965 • http://lego.cuusoo.com/ideas/view/18645 • http://footage.shutterstock.com/clip-1240201-stock-footage-radar-background-of-an-incoming-aircraft-hd.html\ • http://www.push-start.co.uk/all/platform/nintendo/nintendo-news/wii/harry-potter-lego-new-character-pictures/ • http://www.infoniac.com/hi-tech/top-10-tech-inventions-made-of-lego.html • http://solidalexei.deviantart.com/art/LEGO-Indiana-Jones-Icon-88849002 • http://www.thedailybrick.co.uk/lego-plate-2-x-8-black.html • http://wii.mmgn.com/Gallery/Lego-Two-Face

36. Pictures Continued…. http://www.calculatorsoup.com/calculators/geometry-solids/tube.php http://aboutus.lego.com/en-us/news-room/2012/june/lego-the-lord-of-the-ring-videogame/ http://www.eurobricks.com/forum/index.php?showtopic=67589 http://identifyingminifigures.com/C/spider-man-minifigures http://www.gorickconstructioninc.com/alfredgorickcoincbrsandgravel/ http://lego.wikia.com/wiki/Iron_Man http://ministryofminifigures.com/lego-super-heroes-minifigure-gallery/ http://minifigtimes.com/iminifigure.com/product_info.php/lego-weapons-captain-america-shield-p-346 http://thehobbit.lego.com/en-us/characters/bilbo/ http://funny-pictures.feedio.net/lego-instructions-krusty-krab-lego-helicopter-instructions/lego.brickinstructions.com*03000*3825*035.jpg/ http://whotalking.com/flickr/Krusty+Krab http://produto.mercadolivre.com.br/MLB-483815177-lego-boneco-jar-jar-star-wars-frete-r500-_JM http://www.youtube.com/watch?v=rpW1oER8I1Y http://commons.wikimedia.org/wiki/File:Cone.png

37. Other Info • http://www.maa.org/pubs/Calc_articles/ma009.pdf • http://www.fredonia.edu/department/math/Methods2008/AMTNYSLessonPlans/Related%20Rates.pdf • http://ed-thelen.org/rocket-eq.html