AP * Calculus Review. Related Rates

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1 AP * Calculus Review Related Rates Student Packet AP* is a trademark of the College Entrance Examation Board. The College Entrance Examation Board was not volved the production of this material. Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 46

2 Page 1 of 11 Session Notes Questions that ask for the calculation of the rate at which one variable changes, based on the rate at which another variable is known to change, are usually called related rates. Solutions are found by writg an equation that relates the variables of the problem then differentiatg them with respect to another variable (usually time). Sce time is rarely a variable the equation that you write, you will have to differentiate implicitly with respect to time. You can recognize that the rates are with respect to time by lookg at the units:, cm, or mi hr. Process 1. Draw a picture. Label constant values and assign variables to thgs that change.. Translate the given formation the problem to calculus-speak. Do the same thg for what you are asked to fd. For example, the rate of change of the area is 15 becomes da m dt = 15. m. Write a formula/equation relatg the variables whose rates of change you seek and the variables whose rates of change you are given. Look to geometry for many formulas. Important: At this stage you may substitute for a quantity that is constant; however, don t freeze your problem by substitutg a number for a quantity that is changg keep variables variable!! 4. Differentiate implicitly with respect to time. Use all differentiation rules that apply. 5. Now you can plug numbers and do calculations. If you round off an answer, use three decimal places. 6. Use complete sentences to answer the question that is asked. Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 47

3 Page of 11 Practice for prelimary steps Translate to calculus-speak. 1. The area of a circle is creasg at a rate of 6. m. The volume of a cone is decreasg at a rate of.. The population of a city is growg at a rate of people day. Translate to words. 1. dc dt. dr cm 40 dt. dv dt 8 m Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 48

5 Page 4 of The radius of a circle is creasg at a constant rate of 0.4 meters per ond. What is the rate of crease the area of the circle at the stant when the circumference is 60 m (A) 0.4 (B) m 4 (C) m 4 (D) m 0 (E) m 40. A beach ball is deflatg at a constant rate of 10 cubic centimeters per ond. When the volume of the ball is 56 cubic centimeters, what is the rate of 4 change of the surface area? ( S 4 r and V r ) (A) cm 80 (B) cm 5 (C) 5 cm (D) 5 cm (E) 0 cm Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 50

6 Page 5 of 11. The height of gra a cyldrical silo is creasg at a constant rate of 4 feet per mute. At what rate is the volume of gra the cylder creasg if the radius of the silo is 10 feet? ( V 4 r h ) (A) 1 5 m (B) 5 m (C) 40 m (D) 400 m (E) 400 m 4. A 15 foot ladder is leang agast a buildg when its base begs to slide away from the buildg at constant rate of feet per mute. How fast is the base of the ladder movg away from the buildg when the base of the ladder is 9 feet from the buildg? (A) 64 9 m (B) m (C) m (D) 8 m (E) 7 m Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 51

7 Page 6 of Bikes A and B are travelg on perpendicular roads. At the same time, bike A is leavg the tertion at a rate of feet per ond and bike B is leavg the tertion at feet per ond. How fast is the distance, feet per ond, between them changg aer 5 onds? (A) 1 5 (B) 1 5 (C) 1 (D) (E) 5 1 6GC. In a rectangle, the length is creasg at constant rate of.0 centimeters per ond, while the width is decreasg at a constant rate of 0.6 centimeters per ond. At the time that the length is centimeters and the width is 0.4, the rate of change of the area is (A) -0.0 (B) (C) (D).448 (E) 5.79 Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 5

8 Page 7 of GC. The volume of a spherical balloon ( V r ) is creasg at a constant rate of 0.78 ches per mute. At the stant when the radius is.0 ches, the radius is creasg at a rate of (A) m (B) m (C) m (D) 6.7 m (E) m 8.GC On a construction site, gravel is delivered and poured to a conical pile. The diameter and height of the cone of gravel are changg a way that the diameter is always times the height. If the delivery truck is set to pour the gravel at a constant rate of cubic feet per mute, how fast is the radius of the pile changg 1 when the height is 4 feet? ( V r h ) (A) m (B) m (C) m (D) m (E) m Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 5

9 Page 8 of A camera is filmg the progress as a daredevil attempts to scale the wall of a skyscraper. The climber is movg vertically at a constant rate of 16 feet per mute, and the camera is 400 feet from the base of the skyscraper. Through how many radians per mute is the camera angle changg when the climber is 00 feet up the buildg? (A) 1 65 (B) 1 0 (C) (D) 1 0 (E) A cube has an edge of 40 feet at t 0, and the edge is decreasg at a constant rate of 4 feet per mute. Aer mutes, the rate of change of the volume cubic feet per mute is (A) 84 (B) 480 (C) 6400 (D) 1,88 (E) 19,00 Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 54

10 Page 9 of 11 Free Response 1 No calculator A square is scribed a circle. The radius of the circle is creasg at a constant rate of 0.8 centimeters per ond. (a) When the side of the square is 4 centimeters, what is the area of the circle? Include units. (b) When the side of the square is 4 centimeters, what is the rate of change the area of the circle? Include units. (c) When the radius of the circle is 5 centimeters, what is the rate of change the shaded area of the region outside the square but side the circle? Include units. Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 55

11 Page 10 of 11 Free Response No calculator A pot, P, with coordates (x, y) is movg counterclockwise on a circle centered at (0, 0) with radius 5. y P (x, y) x (a) When P is at (4, ) dx dt is. What is dy dt? (b) What are the signs of dx dt and dy quadrant? Justify your answer. dt (c) Consider the triangle with vertices (0, 0), (x, 0) and (x, y). Usg formation from part (a), fd the rate at which the area of this triangle is changg when P is at (4, ). (d) In the triangle from part (c), is the angle formed by the hypotenuse and the horizontal leg. When P is at (4, ), fd the rate of change of. Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 56

12 Page 11 of 11 Free Response Calculator Allowed Water is poured to a conical tank that is 4 feet tall and has a diameter at the top of 0 1 feet. ( V r h ) (a) Write the formula for the volume of the cone of water terms of h, the height of the water the tank. (b) When the volume of the water is creasg at.4 cubic feet per mute and the height of the water is feet, at what rate is the height of the water changg? (c) The radius of the surface of the water the tank is creasg at 0.75 feet per mute. At what rate is the area of the surface changg when the radius is 4. feet? Copyright 008 Layg the Foundation, Inc., Dallas, TX. All rights reserved. Visit: 57

Related Rates. AP Calculus Review. Teacher Packet. Advanced Programs Division

Related Rates. AP Calculus Review. Teacher Packet. Advanced Programs Division Advanced Programs Division * AP Calculus Review Related Rates Teacher Packet Advanced Placement and AP are registen:d trademark. of the College Entrance Examination Board. The College Board _ not involved