CalcAB
Chp 46B Related Rates, Fixed Amounts and Trigonometry
Notes
1.
Ex(right triangle, Pythagorean theorem)
—
An airplane is flying in a flight path that will take it directly over a radar
tracking station.
The distance (between the plane & station) is decreasing at a
rate of 400 miles per hour when the plane is 10 miles from the station.
What is
the speed of the plane?
Help getting started and some tips
Organize the language
—
the path of the plane is the horizontal (dotted line which is parallel to the ground)
(not the hypotenuse, that would be a crash into the radar station)
“
distance is decreasing at a rate of 400 mph
”
dd/dt = 400 mph
(could make negative)
speed is along the path of the plane, which is the horizontal
“
What is the speed of the plane?
”
dx/dt = ?? mph
“
when plane is 10 miles from station
”
d = 10 mi.
Save this value to use with the derivative later.
An important aspect of each problem to look for is what actually changes and what does not.
The height of the plane DOES NOT CHANGE; the plane stays on the horizontal path above the
ground.
The height is a FIXED amount.
Because the height does not change, you have two
options on how to set up the equation.
Need an equation.
The problem involves a
right triangle and three sides
,
use Pythagorean theorem
.
Option #1:
When making the equation
do not use a variable for height; instead use height=6
(a
leg of a right triangle).
Option #2:
When making the equation
use a variable for height
, say h, (one of the legs of a right
triangle),
then use
dh/dt = 0 mph
.
dh/dt=
0
because there is
no
change in height with respect to time.
Differentiate your equation with respect to hours (dt).
When you plug in known amounts
d=10 and
dd/dt=400
to solve for
dx/dt, you will
notice you need x
.
Use
d=10 and
h=6 and Pythagorean theorem to figure out x.
Answer:
dx/dt = 500 mph
6
mi
x
d
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2.
Ex(volume, cylinder)
—
From previous ASN
Pg 251 #15
The mechanics at Lincoln Automotive are reboring a sixinch deep cylinder to fit a new piston.
The machine they are using increases the cylinder’s radius one
thousandths of an inch every
three minutes.
How rapidly is the cylinder volume increasing when the bore (diameter) is 3.8
inches?
Help getting started and some tips
Picture is not provided with the problem.
The radius that is changing is the radius
of the inner core, not the radius for the whole cylinder.
As in the previous
example, you need to observe what parts are changing and what stay the same
measure.
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 Spring '10
 KNOPF
 Trigonometry, Pythagorean Theorem

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