related rates greg kelly, hanford high school, richland, washington
TRANSCRIPT
Related Rates
Greg Kelly, Hanford High School, Richland, Washington
Related Rates Problems
• All variables are differentiated with respect to TIME
• How quickly?• At what rate?• How fast is the _____________ changing?• Rate of change of ____________?• Write a general equation relating the variables,
then differentiate with respect to time.
Related Rates Problems
A ladder slides down the wall of a building as shown
y
x
How fast the height is changingdy
dt
How fast the distance from the base of the wall is changing
dx
dt
Suppose you were told that the bottom of a 13 foot tall ladder was 5 meters from the wall and moving away from the wall at 8 mps.
How fast is the top of the ladder moving down?
The ladder, the ground, and the wall make up a triangle
x
y13
The rate that the distance from the wall is changing
dxdt
8
The rate that the top of the ladder is falling
dydt
This is what we need to find!
x
y 13
What is the relationship between the three sides of the triangle?
2 2 169x y
2 2 0dx dyx ydt dt
But when x=5, y=12, and 8dx
dt
2 5 8 2 12 0dy
dt
3
10
dy
dt fps
WHY?
Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec.(Possible if the sphere is a soap bubble or a balloon.)
310cm
dV
The change in radius per time
The change in volume per time
dV
dt
dr
dt
0.1 / seccm
Now we need a formula that relates the volume and the radius of a sphere
34
3V r
24dV dr
rdt dt
2 cm4 10cm 0.1
sec
dV
dt
3cm
40sec
dV
dt
The sphere is growing at a rate of . 340 cm / sec
The formula for the volume of a sphere is
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
L3
sec
dV
dt
3cm3000
sec
Finddh
dt2V r h
2dV dhr
dt dt (r is a constant.)
32cm
3000sec
dhrdt
2
3000 cm
sec
dh
dt r
(We need a formula to relate V and h. )
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.
Hot Air Balloon Problem:
Given:4
rad
0.14min
d
dt
How fast is the balloon rising?
Finddh
dt
tan500
h
2 1sec
500
d dh
dt dt
2
1sec 0.14
4 500
dh
dt
h
500ft
Hot Air Balloon Problem:
Given:4
rad
0.14min
d
dt
How fast is the balloon rising?
Finddh
dt
tan500
h
2 1sec
500
d dh
dt dt
2
1sec 0.14
4 500
dh
dt
h
500ft
2
2 0.14 500dh
dt
1
12
4
sec 24
ft140
min
dh
dt
4x
3y
B
A
5z
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?
r t d 1
40 410
130 3
10
2 2 23 4 z 29 16 z
225 z5 z
4x
3y
30dy
dt
40dx
dt
B
A
5z
Truck Problem:
How fast is the distance between the trucks changing 6 minutes later?
r t d 1
40 410
130 3
10
2 2 23 4 z 29 16 z
225 z5 z
2 2 2x y z
2 2 2dx dy dzx y zdt dt dt
4 40 3 30 5dz
dt
250 5dz
dt
50dz
dt
miles50
hour
Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.