Related RatesPeyam Ryan TabrizianTuesday, July 19th, 2011How to solve related rates problems1) Draw a picture!, labeling a couple of variables. HOWEVER do notput any numbers on your picture, except for constants! Otherwise you’llget confused later on2) Figure out what you ultimately want to calculate, and don’t lose track of it3) Find an equation relating your variables4) Differentiate your equation using the chain rule/implicit differentiation.5) NOW plug in all the numbers you know! Sometimes, you might need tocalculate a number of ’missing variables’. Here an extra picture as in 1),but with all the numbers plugged in, might be useful6) 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 =π3r2h• Volume of a cylinder: V = πr2h• Volume of a ball: V =43πr31Problem 1If z = x2+ y2, finddzdtwhen x = 3, y = 4,dxdt= 3, anddydt= −2.Problem 2[3.9.19] The altitude of a triangle is increasing at a rate of 1 cm/min while thearea of the triangle is increasing at a rate of 2 cm2/min. At what rate is thebase 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 6mi/h and Car B travels East at a rate of 2.5 mi/h. At what rate is the distancebetween the two cars increasing 2 hours later?Problem 4A ladder 10 ft long rests against a vertical wall. If the bottom of the ladderslides away from the wall at a rate of 1 ft/s, how fast is the top of the laddersliding down the wall when the bottom of the ladder is 6 feet from the wall?Problem 5A ladder 10 feet long rests against a vertical wall. The bottom of the ladderslides away from the wall at a rate of 1 ft/s. How fast is the angle between theladder 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 theladder and the ground.Problem 6Assume Peyam’s happiness is given by H = L2√M, where L is the number ofutils (happiness points) due to teaching Math 1A lectures, and M is the numberof utils due to holding office hours. If currently L = 10 and is increasing by4 utils/day and M = 100 and is decreasing by 10 utils/day, is Peyam gettinghappier or sadder now, and at what rate?Problem 7A cylindrical gob of goo is undergoing a transformation in which its height isdecreasing at a rate of 1 cm/s while its volume is decreasing at the rate of 2πcm3/s. (It retains its cylindrical shape while all of this is happening). If, at agiven instant, its volume is 24π cm3and its height is 6 cm, determine whetherits radius is increasing or decreasing at that instant, and at what
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