Related Rates Peyam Ryan Tabrizian Tuesday July 19th 2011 How to solve related rates problems 1 Draw a picture labeling a couple of variables HOWEVER do not put any numbers on your picture except for constants Otherwise you ll get confused later on 2 Figure out what you ultimately want to calculate and don t lose track of it 3 Find an equation relating your variables 4 Differentiate your equation using the chain rule implicit differentiation 5 NOW plug in all the numbers you know Sometimes you might need to calculate a number of missing variables Here an extra picture as in 1 but with all the numbers plugged in might be useful 6 Solve for whatever you were looking for in 2 List of tricks Pythagorean theorem Definition of sin and cos Formulas for areas and or volumes Volume of a cone V 2 3r h Volume of a cylinder V r2 h Volume of a ball V 34 r3 1 Problem 1 If z x2 y 2 find dz dt when x 3 y 4 dx dt 3 and dy dt 2 Problem 2 3 9 19 The altitude of a triangle is increasing at a rate of 1 cm min while the area of the triangle is increasing at a rate of 2 cm2 min At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100cm2 Problem 3 3 9 15 Two cars start at the same point Car A travels North at a rate of 6 mi h and Car B travels East at a rate of 2 5 mi h At what rate is the distance between the two cars increasing 2 hours later Problem 4 A ladder 10 ft long rests against a vertical wall If the bottom of the ladder slides away from the wall at a rate of 1 ft s how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall Problem 5 A ladder 10 feet long rests against a vertical wall The bottom of the ladder slides away from the wall at a rate of 1 ft s How fast is the angle between the ladder and the wall changing when the bottom is 6 feet from the wall Note Careful On your homework they ask you about the angle btw the ladder and the ground Problem 6 Assume Peyam s happiness is given by H L2 M where L is the number of utils happiness points due to teaching Math 1A lectures and M is the number of utils due to holding office hours If currently L 10 and is increasing by 4 utils day and M 100 and is decreasing by 10 utils day is Peyam getting happier or sadder now and at what rate Problem 7 A cylindrical gob of goo is undergoing a transformation in which its height is decreasing at a rate of 1 cm s while its volume is decreasing at the rate of 2 cm3 s It retains its cylindrical shape while all of this is happening If at a given instant its volume is 24 cm3 and its height is 6 cm determine whether its radius is increasing or decreasing at that instant and at what rate 2
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