The volume of a sphere and the radius of the same sphere are related by the formula:

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1 Related Rates Today is a day in which we explore the behavior of derivatives rather than trying to get new formulas for derivatives. Example Let s ask the following question: Suppose that you are filling a water balloon up from a tap pouring water at a rate of one cubic centimeter (also known as a milliliter) per second. This tells you how fast the volume is changing. Let V (t) = volume after t seconds. dv dt = The volume of a sphere and the radius of the same sphere are related by the formula: ( ) V = Question Can we use the information given thusfar to determine the rate of change of the radius: dr at the moment when the radius of the balloon is 10 cm? dt Differentiating all of the quantities in ( ) with respect to t gives us: = Next plug in that dv dt to get the equation = and that at the moment of interest V = = Solve this for dr dt. This philosophy is that of related rates: If you have a relationship between quantities A and B and you know how fast A or B is changing, the you probably know how fast the other is changing. As a step by step process: Find a formula which relates the quantities you are interested in. Take the derivative of this formula. Plug in the values which are known. Solve for the quantity you would like to know. 1

2 2 (1) You throw a stone in a lake. A ripple expands out from the center forming a circle whose radius is r(t) = 2 t. What is the derivative of the area in this circle with respect to time? Step 1: Find a relationship between area and radius height = h (2) A five foot ladder leans against a vertical wall. A jerk begins to pull the base of the ladder away from the wall at a rate of 1 foot per second. When the ladder s base is three feet from the base of the wall, how fast is the top of the ladder falling? 5 ft. base = b Step 1: Find a relationship between the location of the bottom and top of the ladder.

3 The remaining examples serve as group work. The rest of today and tomorrow will be spent with you working these out. 3 (3) You have a cup whose shape is an upside down right cone with radius 2 inches and height 4 inches. You begin pouring water into the cup at a rate of 2 cubic inches per second. How fast is the height of the waterline in the cup increasing once there is one cubic inch in the cup height = h radius = r Step 1: Find a relationship between the volume, height, and radius. Translate this into a relationship between volume and radius only. Step 1a: The radius and height are related. Think about the slope of the wall of the cup. Step 1b: Volume is given in terms of radius and height. Use (1a) to get volume in terms of height only.

4 4 (4) While riding in the bus you watch a tree 100 meters away from the road go by. The bus is going 15 meters per second. At the moment the line between you and the tree is perpendicular to the road, how fast is your head spinning (in radians per second)? θ you 100 meters tree y Step 1: Find a relationship between the angle of rotation and the location of the tree relative to you. Think about cotangent (5) While riding in the bus you watch a post 3 meters away from the road go by. The bus is going 15 meters per second. At the moment the line between you and the post is perpendicular to the road, how fast is your head spinning (in radians per second)? θ you 3 meters tree y Step 1: Find a relationship between the angle of rotation and the location of the post relative to you. (6) How much faster is your head spinning this time?

5 5 (7) You have a rectangle whose edge lengths are changing. At some instant you observe that h = 1 meter and w = 2 meters. Measuring the rate of change at that same instant, you see that the height is growing at a rate of 1 meter per second and the width is shrinking at the same rate. Compute the rate of change of the area of the box and of the length of its diagonal. The area: Step 1: Find a relationship between w, h, and A. h w Step 3: Plug in the given values. (Some are negative be careful!) The diagonal: d h w Step 1: Find a relationship between w, h, and d. Step 3: Plug in the given values. (Some are negative be careful!)