The Misadventures of Gale the Birthday Clown Related Rates Problems in 5 Easy Steps Gale dreamed of being an actor. He became a birthday clown. 1. Little Billy’s parents hire Gale the birthday clown to perform at Billy’s 5th birthday party. While the children arrive, Gale is blowing up a fun spherical shaped balloon at a constant rate of 100 cubic inches per minute. How fast is the radius growing when the sphere contains 50 cubic inches of birthday cake smelling clown breath? 1. Read the problem three times. The first time, get a general sense of the question. Second time. highlight what the problem asks for. Third time, search for keywords and mark-up the text. 2. Draw a diagram and label the variables. List the information you know and be sure to note what elements are fixed. Identify the “freeze frame” moment. 3. Establish the relationship between all of the variables using an equation. This is called the relating equation. 4. We are focused on time, so take the derivative both sides of the equation with respect to time (You will need to use the chain rule/implicit differentiation) 5. Plug in the information given and solve for the unknown variable or rate. 2. Gale popped most of the balloons. Annoyed with their clown, Billy’s parents have assigned him the task of filling up the birthday boy’s new pool. Gale reads on the box that the cylindrical pool has a radius of 5 feet. Gale is pumping at a rate such that the height of the water changes at a constant rate of 0.1 ft per minute. Water is seeping out a hole in the bottom of the pool at a rate of 3 cubic feet per minute because Gale accidentally walked over it in his giant clown shoes. At what rate is Gale pumping water into the pool? 3. Gale is tired and a touch light headed after blowing up all those balloons and the kiddie pool. He waddles inside and opens up a nice refreshing can of soda but accidentally knocks it over on the table where Billy’s friends have stacked all the gifts. The soda spills out at a constant rate of 7 ml/second and forms a cylindrical spill. Gale can tell that when the radius of the spill is 5 cm and the height is 2 mm the height grows at a rate of 0.1 mm/sec. If the radius is growing faster than 1 cm per second, Gale will not be able to save the gifts. Will the presents be ruined? 4. The Obligatory Ladder Theorem When working with real world applications of advanced mathematics it is obligatory to involve a ladder at some point, preferably one leaning against a wall. Gale is trying to escape from the party from an upstairs window, since he just got paid and doesn’t want to blow up any more balloons or pools. Unfortunately, the 20 foot extension ladder he leaned against the wall when he arrived is now slipping down the side of the house. He notices that the bottom is sliding away from the house at a constant rate of 2 ft/sec. How quickly is the top of the ladder sliding down the wall at the exact moment the base of the ladder is 12 feet away from the house? Should Gale jump for it? How many more balloons will Gale have to blow up if his escape fails? 5. Gale lets all the balloons he blew up loose outside. They get caught up in some electrical wires. Billy’s dad forces Gale to climb the telephone pole to get them back. Dad is standing still 20 feet back from the base of the pole because he doesn’t want a clown to fall on him. He notices that as Gale climbs the angle of elevation from him to Gale is changes at a rate of 0.046 radians per minute when the angle of elevation is 0.75 radians? At what rate is Gale climbing? Why is Gale moving so slowly? 6. Gale is climbing so slowing because he is convinced that he is going to be electrocuted once he gets to the top of the pole. He is thinking about electricity and in particular resistors… A resistor is a two-terminal electronic component that produces a voltage across its terminals that is proportional to the electric current through it in accordance with Ohm's law: Resistors are elements of electrical networks and electronic circuits and are ubiquitous in most electronic equipment. Practical resistors can be made of various compounds and films, as well as resistance wire (wire made of a high-resistivity alloy, such as nickel-chrome). The primary characteristics of a resistor are the resistance, the tolerance, the maximum working voltage and the power rating. Other characteristics include temperature coefficient, noise, and inductance. He is considering two resistors connected in parallel with resistances and measures in ohms ( ). The total resistance, R, is then given by, Suppose that is increasing at a rate of 0.4 is R changing when and /min and is decreasing at a rate of 0.7 ? Gale will be electrocuted if the rate of increase is greater than 0.1 /min . Does he survive? /min. At what rate 7. Bet you saw this coming a mile away. Gale is now frozen in fear at the top of the pole. The children have a plan. They push the trampoline over so it is directly below Gale and convince him that it is safe for him to drop down. Actually, they want to get a video of Gale falling from the electrical pole and getting hit with a pie on his way down to post on YouTube. The birthday boy sets up his new video camera 15 feet back from the pole and focuses on Gale who has now reached a height of 25 feet. The kids are ready with the pies and will hit him when he is 8 feet from the ground. The camera can’t stay in focus if the angle is changing any faster than . Does Billy get the shot?
