4.6 Related Rates

Suppose a ladder leans against a wall as the bottom is pulled away at a constant velocity. How fast does the top of the ladder move? What is interesting and perhaps surprising is that the top and bottom of the ladder travel at different speeds. Notice that the bottom travels the same distance over each time interval, but the top travels farther during the second time interval than the first…and even more during the second time interval. The top is speeding up!

With related rate problems, the idea is to compute the ROC of one quantity in terms of the ROC of another quantity. Find an equation that relates both quantities and differentiate with respect to time.

A 5 meter ladder leans against a wall. The bottom of the ladder is 1 A 5 meter ladder leans against a wall. The bottom of the ladder is 1.5 m from the wall at t = 0 and slides away from the wall at a rate of 0.8 m/s. Find the velocity of the top of the ladder at time t = 1. The second step in any related-rates problems is to choose variables for the relevant quantities. Let x = distance from bottom of ladder to wall Let h or y = distance from top of ladder to ground.

A 5 meter ladder leans against a wall. The bottom of the ladder is 1 A 5 meter ladder leans against a wall. The bottom of the ladder is 1.5 m from the wall at t = 0 and slides away from the wall at a rate of 0.8 m/s. Find the velocity of the top of the ladder at time t = 1. What do we know? Both are functions of time. We know that and x = 1.5 m when t = 0 What are we trying to find? dh/dt We need an equation relating h and x. Any ideas? x2 + h2 = 52

To calculate dh/dt, differentiate both sides with respect to t. X2 + y2 = 52 To calculate dh/dt, differentiate both sides with respect to t.

To apply the formula, we must find x and h at time t To apply the formula, we must find x and h at time t. Since the bottom slides away at 0.8 m/s and x at t = 0 is 1.5, we have x at t = 1 must be 2.3. How can we find h at t = 1? That’s right…the Pythagorean Theorem.

Exploration: Related Rates (4-9a Key Curriculum Press)

Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

Consider a bathtub whose base is 18 square feet. How fast is the water level rising if water is filling the tub at a rate of 0.7 cubic feet a min? At what rate is water pouring into the tub if the water rises at a rate of 0.8 feet per min?

The radius of a circular oil slick expands at a rate of 2 m per minute. How fast is the area of the oil slick increasing when the radius is 25 m? If the radius is 0 at time t = 0, how fast is the area increasing after 3 min?

Assume that the radius of a sphere is expanding at a rate of 30 cm/min Assume that the radius of a sphere is expanding at a rate of 30 cm/min. The volume of a sphere is: and its surface area is . Find the change of volume with respect to time when r = 15 cm. Find the change of volume with respect to time at t = 2 min, assuming that r = 0 at time t = 0. Find the change in SA with respect to time when r = 40 cm. Find the change in SA with respect to time at t = 2 min, assuming that r = 10 cm at t = 0.

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?

Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? B A

p Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? B A p