Unit 3: Applications of the Derivative A.P. Calculus (AB) – Mr. Lajoie – Blocks E and F Guided Notes – November 2013 PART TWO Related Rates (Chapter 3 Section 11)

3.11 Related Rates Related rates describe problems that involve two or more variables that are changing with respect to time. The rate at which one variable is changing with respect to time is related to the rate at which another variable is changing with respect to time, and vice versa.

Guidelines for Solving Related Rates Problems 1) Read the problem carefully to allow for understanding. 2) Identify ALL variables, especially the variable whose rate of change you seek and the variable(s) whose rate of change you know. 3) Draw a diagram. Label all important aspects. Distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start.

Guidelines for Solving Related Rates Problems (cont.) 4) Write an equation relating the variables in the diagram, particularly those whose rate of change you seek with those whose rates of change you know. 5) Differentiate both sides of (4) with respect to time, t. 6) Substitute all values whose quantities and/or rates of change you know. Always substitute after differentiating!!! Solve for the unknown rate of change. 7) Think about your answer. Does it make sense?

Example 1 My cylindrical Starbucks coffee mug has a 3 inch diameter and is 4 inches tall. How many cubic inches of coffee can it hold? Coffee pours into the mug at a constant rate from my Keurig brewer every morning, and it is filled right to the top of the mug. It takes 90 seconds to fill. What is the rate of change of volume of coffee in my mug with respect to time? Determine the rate of change of the height of the coffee in the mug with respect to time 30 seconds into a brew.

Example 2 A) Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft/min. Find the rate of change of the radius when the radius is 2 feet. B) The radius of a balloon is increasing at a rate of 2 inches per minute. At what rate (in cubic inches per minute) is the volume increasing when the surface area of the sphere is 𝟗𝝅 square inches?

Example 3 Assume that the radius 𝑟 of a sphere is expanding at a rate of 30 cm/min. Determine the rate of change of the volume with respect to time at 𝑡=2 min, assuming that 𝑟=0 at 𝑡=0. Determine the rate of change of the surface area with respect to time at 𝑡=2 min, assuming that 𝑟=10 at 𝑡=0.

Example 4 A) A pebble is thrown into a pond forming ripples in the form of concentric circles whose radius increases at the rate of 4 inches per second. How fast is the area of the ripple changing 3 seconds after the pebble drops in the pond? B) The circumference of a circle is increasing at a rate of 2𝜋 5 inches per minute. When the circumference is 10𝜋 inches, how fast is the area of the circle increasing in square inches per minute?

Example 5 The diagonal of a square is increasing at a rate of 3 inches per minute. When the area is 18 square inches, how fast (in inches per minute) is the perimeter increasing?

Example 6 At what rate is the diagonal of a cube increasing if its edges are increasing at a rate of 2 cm/s?

Example 7 A 10 foot ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the top of the ladder sliding down the wall at 𝑡=6 𝑠?

Example 8 The volume of a cone of radius 𝑟 and height ℎ is given by 𝑉= 1 3 𝜋 𝑟 2 ℎ. If the radius and the height both increase at a constant rate of 1 2 centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?

Example 9 An inverted conical container has a diameter of 42 inches and a depth of 15 inches. If water is flowing out of the vertex of the container at a rate of 35𝜋 cubic inches per second, how fast is the depth of the water dropping when the height is 5 inches?

Example 10 A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person’s shadow is lengthening?

Example 11

Example 12 Two cars are traveling along perpendicular roads, car 𝐴 at 40 mph, car 𝐵 at 60 mph. At noon, when car 𝐴 reaches the intersection, car 𝐵 is 90 miles away, and moving toward it. At 1pm, what is the rate (in mph) at which the distance between the cars is changing?

Example 13 A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 radians per minute. How fast is the balloon rising at that moment?

Example 14 A particle is traveling along the curve 𝒙 𝟐 −𝒙𝒚+ 𝒚 𝟐 =𝟕. At the moment when the particle is at the point 2, 3 , its 𝑥- coordinate is increasing at a rate of 5 units/minute. What is the rate of change of the 𝑦-coordinate at this moment?

AP SAMPLE!

CHALLENGE! The minute hand of a clock is 8 cm long, and the hour hand is 5 cm long. How fast is the distance between the tips of the hands changing at 3 o’clock?