RELATED RATES

Nick BignerTJ HowardThao Vu

Definition of Related Rates

Related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.

Steps to solving a related rates problem

Step 1: draw a diagram related to the problem and label accordingly

Step 2: specify in mathematical for the rate of change you are looking for and record all given information

Step 3: find an equation involving the variable whose rate of change is to be found

Step 4: differentiate with respect to time (t)

Step 5: state the final answer in correct form, and specifying the units that you are using

World application Some real world

applications of related rates includes predicting collisions, rate at which

liquid drains, travel, satellites orbiting the earth, increasing the volume of an object,

mars expeditions, pretty much anything Nasa or any physicists for that matter do, and much

much more!

Example Tugboat Problem: A tugboat moves a ship up

to the dock by pushing its stern at a rate of 3m/sec. The ship is 200m long. Its bow remains in contact with the dock and its stern remains in contact with the pier. At what rate is the bow moving along the dock when the stern is 120m from the dock?

Work

Dx/dt= ?X= ?

Dy/dy= 3m/sY= 120mDz/dt= 0Z= 200m

Work

X2 + y2 = z2

2002 – 1202 = x2

X= 160m

Work

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

(160)(dx/dt) + (120)(-3) = (200)(0)

(160)(dx/dt) = 360

Answer

Dx/dt = 2.25m/s

Example Slag Heap Problem: Slag left over from the

manufacturing of cement is being piled outside the cement plant. The resulting slag heap is a cone whose elements make an angle of 40 degrees with the horizontal. Environmentalists measure the circumference of the heap one day, finding it to be 3000ft, and increasing at 7ft/day. About how fast is the cement plant generating slag?

WORK

WORK

V=1/3(п)(r2)(h)

C=2(п)(r2)

R=(c)/2(п)

H=(r)tan40°

V=1/3(п)((c)/2(п))3 tan40°

WORK

dV/dt = (c2)(dc/dt)(tan40°)/(8п2)

dV/dt = (c2)(7)(tan40°)/(8п2)

dV/dt = (30002)(7)(tan40°)/(8п2)

Answer

dV/dt = 669,521.22 ft 3 /day

EXAMPLE Rectangle Problem

The length of a rectangle is increasing at 3ft/min, and the width is decreasing at 2ft/min. When the length is 50ft and the width is 20ft, is the area of the rectangle increasing or decreasing? At what rate?

WORKX=50

Dx/dt=3Y=20

Dy/dt=-2A=1000Da/dt=?

Work

LxW=A

(20)(50)=1000

X(dy/dt) + Y(dx/dt) =dA/dt

dA/dt = 3(20) + -2(50)

Answer

decreasing at a rate of

dA/dt = 40ft/min

FRQ Barn Ladder Problem:

A ladder to the loft in a barn is arranged so that it can be pushed up against the wall when it is not in use. The top of the ladder slides in a track on the wall, and the bottom is free to roll across the floor on wheels. To make the ladder easier to move, a counterweight is attached to the top of the ladder by a rope over a pulley. As the ladder goes away from the wall, the counterweight goes up and vice versa.

FRQ Questions

A. The ladder is 20 ft long. Write an equation expressing the velocity the counterweight moves as a function of the distance the bottom of the ladder is from the wall and the velocity the bottom of the ladder moves away from the wall.

B. Find the velocity of the counterweight when the bottom is 4ft from the wall, and is bring pushed toward the wall at 3 ft/s.

C. If the ladder is allowed to drop all the way to the floor with the bottom moving at 2ft/s and the top remains in contact with the wall, how fast is the counterweight moving when the top just hits the floor?

Work A

202 = x2 + y2

(20)(0) = x(dx/dt) + y (dy/dt)

Dy/dt = -(x/y)(dx/dt)

V = (x)/((202)-(x2))1/2 (dx/dt)

WORK B

v = (4)/(384)1/2 (-3)

V = -.0623 m/s

WORK C

X = 20

Dx/dt = 2

V = undefined

Try Me!

Bacteria Spreading Problem: Bacteria are growing in a circular colony one bacterium thick. The bacteria are growing at a constant rate, thus making the area of the colony increase at a constant rate of 12 mm^2/hr. How fast is r changing when it equals 3 mm.

Answer

dr/dt = 6/πr

Try Me! An airplane is flying 600 mi/hr on a

horizontal path that will take it directly over an observer. The airplane is 7mi high. How fast is z changing when x is 10 mi?

Answer•X= 10•Dz/dt= -491.539 which mean that the distance is decreasing at about 492 miles/hour.

Try Me!

Cone of Light problem: A spotlight shines on the wall, forming a cone of light in the air. The light is being moved closer to the wall, making the cone’s altitude decrease at 6ft/ min. At the same time, the light is being refocused, making the radius increase at 7ft/ min. At the instant when the altitude is 3 ft and the radius is 8 ft, is the volume of the cone increasing or decreasing? How fast?

Answer

Volume is decreasing at about 50.3 ft 3 /

min.

Darth Vader Problem!Darth Vader’s spaceship is approaching the origin along the positive y-axis at 50km/sec. Meanwhile, his daughter Ella’s spaceship is moving away from the origin along the positive x-axis at 80 km/sec. When Darth is at y = 1200km and Ella is at x = 500km, is the distance between them increasing or decreasing? At what rate?

Answer

The distance is decreasing at about 15.4m/s

Luke and Leia Problem!

Luke and Leia are trapped inside a trash compactor on the Death Star. The side walls are moving apart at 0.1 m/sec, but the end walls are moving together at 0.3 m/sec. The volume of liquid inside the compactor is 20 cubic meters, a constant.

ANSWER

The depth is increasing at 0.02m/s

Work Cited

Foerster, Paul. Calculus Concepts and Applications. Key Curriculum Press, 1998. Print.

© Nicholas Bigner, TJ Howard, and Thao Vu