Calculus 1000A/B Lecture 26: Calculus 1000 A -Lecture 26- Section 3.9- Related Rates

1. Translate the following problem into a mathematical statement using the given variables: The volume Vof a melting snowball is decreasing at a rate of 4 time, in seconds, since we started our investigation. At what rate is the radius r of the snowball decreasing with respect to time t at the instant when the radius γ of the snow ball is 3 cm ? (The volume V of the sphere in terms of the radius r is V =-TT*) 3 cm . Let the variable t represent the sec 4 3 dV 3 dr dv dr dr . Find-when r=3cm. B.--- 4-Find when r=3cm sec dV sec dt dV = 4cm3. Find ar wh en r 3 cm. ー· Find-when sec 3 cm. dt sec dt dt E. none of these. 2. Answer the question that was set up in problem 1. That is, find the rate at which the radius r of the snowball is decreasing with respect to time t at the instant when the radius r of the snow ball is 3 cm. Round your answer to three decimal place accuracy.

Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013

Exercises 2.2 Terms and Concepts 1 What is the instantaneous rate of change of position In Exercises 15-18, graphs of functions Fx) and gx) ar iven. Identify which function is the derivative of the other. called? 2. Given a function y fix, in your own words describe to find the units of f(x). 3. What functions have a constant rate of change? Problems 4. Given/(5)= 10 andr(S)-2, approximate/(6). 5. Grven P(100) = -67 and P(300) 5·approximate PI110) 6. Given z(25)-187 andz(25) 17, approximate z (20) 7. Knowing02 andf(10)-5 and the methods de- scribed in this section, which approximation is ikely to be most accurate: 10.1. f(11), orfi 201 ? Explain your rea- soning 8. Given A(7)-26 and 8)- 22, approximatef'(7). 9. Given H(0)-17 and H(2)-29, approximate ㎡(2). 17, 10. Let Wx) measure the volume, in decibels, measured inside a restaurant with xcustomers. Whatare the units of V()? 11. Let vit) measure the velocity, in ft/s, of a car moving in a straight line t seconds after starting. What are the units of 18. 12. The height M, infeet, of a river is recorded t hours after mid night, April 1 What are the units of H()? 13. Pis the profit, in thouiands of doliars, of producing and sell ing c cars Review al What are the units of P(c) b) what is likely true of PiO) In Exercises19-20, se the definition to compute the deriv tives of the followinge functions 14. T is the temperature in degrees Fahrenheit, h hours after Sidney, NE. 19. fx) midnight on July 4 20. fx)-(-z) In Exercises 21-22, mumerically approximate the value c f (x) at the indicated x value. (a) What are the units of T"(h)P (b) IsT (8)Skely greater than or less than O? Why? e) s T(8) likely greater than or less than 0? Why?