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Related Rates. Lesson 6.5. General vs. Specific. Note the contrast … General situation properties true at every instant of time Specific situation properties true only at a particular instant of time We will consider a rock dropped into a pond … generating an expanding ripple.
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Related Rates Lesson 6.5
General vs. Specific • Note the contrast … • General situation • properties true at every instant of time • Specific situation • properties true only at a particular instant of time • We will consider a rock dropped into a pond … generating an expanding ripple
Expanding Ripple • At the point in time whenr = 8 • radius is increasingat 3 in/sec • That is we are given • We seek the rate that the area is changing at that specific time • We want to know r = 8
r A Solution Strategy • Draw a figure • label with variables • do NOT assign exact values unless they never change in the problem • Find formulas that relate the variables
Solution Strategy • Differentiate the equation with respect to time • Substitute in the given information
Example • Given • Find when x = 3Note: we must differentiate implicitly with respect to t
Example • Now substitute in the things we know • x = 3 • Find other values we need • when x = 3, 32 + y2 = 25 and y = 4
Example • Result
Particle on a Parabola • Consider a particle movingon a parabola y2 = 4x at (1,-2) • Its horizontal velocity(rate of changeof x) is 3ft/sec • What is the vertical velocity,the rate of change of y? •
Particle on a Parabola • Differentiate the original equation implicitly with respect to t • Substitute in the values known • Solve for dy/dt
Draining Water Tank • Radius = 20, Height = 40 • The flow rate = 80 gallons/min • What is the rate of change of the radius when the height = 12?
Draining Water Tank • At this point in timethe height is fixed • Differentiate implicitly with respect to t, • Substitute in known values • Solve for dr/dt
Assignment • Lesson 6.5 • Page 409 • Exercises 1 – 27 odd