Sect. 2.6 Related Rates

Warm Up If the radius of a spherical balloon changes from 10 cm to 10.1cm, how much does the volume change? How about from 10.4 cm 10.5?

Same problem, using Calculus (differentials): The volume when r = 10, would change by approximately if the radius increases 0.1 cm. The volume when r = 10.4, would change by approximately if the radius increases 0.1 cm.

Now, suppose that the radius is changing at an instantaneous rate of 0 Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec. How fast is the volume of the balloon changing as time increases? The volume of the balloon grows at different rates as time passes. Each rate is dependent upon the radius of the balloon at time t and the rate of change in radius.

Related Rates If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other and the amount of time that has passed. In real-life, we need to be able to solve related rates questions such as: How fast is the ladder slipping down a wall? How fast is the shadow moving? How quickly is the angle decreasing? How fast is the height changing in the water tank? How fast is the area/volume changing? In this section, we will learn how to compute the rate of change of one quantity in terms of that of another.

Steps for Related Rates Problems 1. Draw and label a picture. Use appropriate variables to represent the values involved. Label all rates of change given and those to be determined, using calculus notation (dV/dt, dr/dt, etc). 4. Write an equation to relate the variables. Substitute any CONSTANT values. Differentiate (by implicit differentiation) with respect to t. Substitute all known values and evaluate. Don’t forget your units.

Back to last example: Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. How fast is its volume changing when the radius has reached 10 cm? First, we need the volume formula: Next, what are we looking for? (Remember: How fast is its volume changing as time increases?) r = 10 cm Then, what do we know? Now what do we do?

Change in distance? m/s Change in area? m2/s Change in volume? m3/s r = 10 cm The sphere is growing at a rate of when r = 10cm. Units: Change in distance? m/s Change in area? m2/s Change in volume? m3/s Also note: If the quantity increases: + rate If the quantity decreases: - rate

You Try… Air is being released from a spherical balloon at the rate of 4.5 in3 per second. Find the rate of change of the radius when the radius is 2 inches. Given: r = 2 in. Equation: Diff. & Solve:

Formulas You May Need To Know

How fast is the ladder falling when it is 10 ft off the ground? Example 2: Tweety is resting in a bird house, 24 ft off the ground. Using a 26 foot ladder, leaned against the pole holding the bird house, Sylvester tries to steal the small yellow bird. Tweety’s bodyguard, Hector the dog, starts pulling the base of the ladder away from the pole at a rate of 2 ft/s. ! How fast is the ladder falling when it is 10 ft off the ground? Not so fast!!! 26 ft 24 ft

How fast is the ladder falling (dy/dt) when it is 10 ft off the ground (y)? Known Values: x Equation: Plug in CONSTANTS: Solve for:

26 ft y L x Equation: Solving for related rates requires the derivative of the equation with respect to time. Solve for:

Equation: Known Values:

Equation: Known Values:

How fast is the ladder falling when it is 10 ft off the ground? Example 2: Tweety is resting in a bird house, 24 feet off the ground. Using a 26 foot ladder, which Sylvester leans against the pole, holding the bird house, Sylvester tries to steal the small yellow bird. Tweety’s bodyguard, Hector the dog, starts pulling the base of the ladder away from the pole at a rate of 2 ft/s. How fast is the ladder falling when it is 10 ft off the ground?

y = 8 by Pythagorean theorem You Try… A ladder, 10 ft tall rests against a wall. If the ladder is sliding away from the bottom of the wall at 1 ft/sec, how fast is the top of the ladder coming down the wall when the bottom is 6 ft from the wall? Find dy/dt 10 y x The ladder is moving down the wall at ¾ ft/sec when it is 6 ft. from the wall. At x = 6, y = 8 by Pythagorean theorem

WARNING! The most common error to be avoided is the premature substitution of the given data, before rather than after implicit differentiation.

Example 3: Batman and the Scooby Doo gang are having lunch together when they both simultaneously receive a call. Batman heads off to Gotham city traveling north at 40 miles per hour. The Scooby gang hops in the mystery machine and heads east at 30 miles an hour. How fast is the distance between them changing 6 minutes later?

The bat mobile travels east at 40 mi/hr. The mystery machine travels north at 30 mi/hr. How fast is the distance between the vehicles changing 6 minutes later? B S

Batman travels north at 40 mi/hr. Scooby travels east at 30 mi/hr. How fast is the distance between the vehicles changing 6 minutes later? B A

You Try… Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

We draw this figure, where C is the intersection of the roads. At a given time t, let x be the distance from car A to C. Let y be the distance from car B to C. Let z be the distance between the cars— where x, y, and z are measured in miles. Given: dx / dt = –50 mi/h and dy / dt = –60 mi/h. The rates are negative because x and y are decreasing. Find dz / dt.

z 2 = x 2 + y 2 When x = 0.3 mi and y = 0.4 mi, the Pythagorean Theorem gives z = 0.5 mi. So, The cars are approaching each other at a rate of 78 mi/h.

Ex 4: Given y = x2 + 3, find dy/dt when x = 1, given that dx/dt = 2. When x = 1 and dx/dt = 2, we have

Ex 5. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When this radius is 4 ft., what rate is the total area A of the disturbed water increasing? Givens: Equation:

r = 4 A 4

You Try… A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 3 in/sec. When this radius is 8 ft., what rate is the total area A of the disturbed water increasing? r = 8

Ex 6: Water is draining from a cylindrical tank of radius 20 cm at 3 liters/second. How fast is the water level dropping? 1 liter = 1000 cm3

You Try…

Ex 7: A fish is reeled in at a rate of 1 foot per second from a bridge 15 ft. above the water. At what rate is the angle between the line and the water changing when there is 25 ft. of line out? z 15 ft.

Given: Find: Equation: x = 25 ft. h = 15 ft.

You Try… A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight? dx / dt = 4 ft/s dθ / dt = ? when x = 15

dx / dt = 4 ft/s dθ / dt = ? when x = 15 When x = 15, the length of the beam is 25. So, cos θ = and The searchlight is rotating at a rate of 0.128 rad/s.

We have three variables 8 cm Ex 8: A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm3/min. Find the rate at which the fluid depth h decreases when h = 5 cm. draining at 2 cm3/min What is the rate of change of the height when the height = 5? 9 cm We have three variables But Wait! Whatever shall we do?

Now we have two variables! 8 cm 9 cm 4 cm r 9 cm h Now we have two variables! Similar triangles

Will this rate increase or decrease as h gets lower? 8 cm 9 cm Will this rate increase or decrease as h gets lower? Problems like this surface often so remember your geometric relationships such as similar triangles, etc.

Show that this is true by comparing 8 cm Show that this is true by comparing both when h = 5 cm and when h = 3 cm. 9 cm As h gets smaller, gets faster because h is in the denominator.

You Try… A cone filter of radius 20 in and height 40 in is draining at a rate of 80 gallons/min. Find the rate at which the radius changes when h = 12 in. draining at 80 gallons/min What is the rate of change of the radius when the height = 12?

cont… At this point in time the height is fixed Differentiate implicitly with respect to t, Substitute in known values Solve for dr/dt

Using similar triangles, the equation is: Example 9 A person 6 ft tall is walking away from a streetlight 20 ft high at the rate of 7 ft/s. At what rate is the length of the person’s shadow increasing? 20 ft 6 ft y x Using similar triangles, the equation is: ft/s

The size of his shadow is reducing at a rate of 10/11 ft/s. You Try… A man 5 ft tall is walking at a rate of 2 ft/s toward a street light 16 ft tall. At what rate is the size of his shadow changing? x y 16 5 The size of his shadow is reducing at a rate of 10/11 ft/s.