Transcript Document 418529

Related Rates and
Applications
Lesson 3.7
1
General vs. Specific
• Note the contrast …
• General situation
– properties true at every instant of time
• Specific situation
– properties true only at a particular instant of time
• We will consider a rock dropped into a pond
… generating an expanding ripple
2
Expanding Ripple
• At the point in time when
r=8
– radius is increasing
at 3 in/sec
– That is we are given
r=8
dr
3
dt
• We seek the rate that the area is changing at
that specific time
– We want to know
dA
dt
View Spreadsheet
demonstration
3
Solution Strategy
1. Draw a figure


label with variables
do NOT assign exact values
unless they never change
in the problem
A
r
2. Find formulas that relate the variables
A r
2
dr
3
dt
4
Solution Strategy
3. Differentiate the equation with respect to
time
dA
dr
 2 r
dt
dt
4. Substitute in the given information
r 8
dr
3
dt
2  8  3  48 in / sec
2
5
Example
• Given
x  y  25
2
2
dx
4
dt
dy
• Find
when x = 3
dt
Note: we must differentiate implicitly
with respect to t
dx
dy
2x  2 y
0
dt
dt
6
Example
• Now substitute in the things we know
–
dx
4
dt
x=3
• Find other values we need
– when x = 3,
32 + y2 = 25
y=4
and
dx
dy
2x  2 y
0
dt
dt
7
Example
dx
dy
2x  2 y
0
dt
dt
• Result
dy
244  24
0
dt
dy 32

 4
dt
8
8
Guidelines for Related-Rate Problems
1. Identify given quantities, quantities to be
determined
•
Make a sketch, label quantities
2. Write equation involving variables
3. Using Chain Rule, implicitly differentiate
both sides of equation with respect to t
4. After step 3, substitute known values, solve
for required rate of change
9
R1
Electricity
• The combined electrical
R
resistance R of R1 and R2
1 1
1
 
connected in parallel is
R R1 R2
given by
• R1 and R2 are increasing at rates of 1 and
1.5 ohms per second respectively.
• At what rate is R changing when R1 = 50
and R2 = 75?
2
10
Draining Water Tank
• Radius = 20, Height = 40
•
1 2
Volume   r h
3
• The flow rate = 80 gallons/min
• What is the rate of change of
the radius when the height = 12?
dV
 80
dt
dr
 ??
dt
11
Draining Water Tank
• At this point in time
the height is fixed
1 2
Volume   r 12
3
• Differentiate implicitly
with respect to t,
dV 1 
dr 


2

r


12


• Substitute in known
dt 3 
dt 
values
• Solve for dr/dt
12
Assignment
• Lesson 3.7
• Page 187
• Exercises 1 – 7 odd, 13 – 27 odd
13