Download 2.6 Related Rates (Day 2) Objective: Find a related rate and use

Miss Battaglia
AP Calculus
Related rate problems
involve finding the
rate at which some
________
variable changes.
For example, when a balloon
is being blown up with air,
radius
both the _____________
and
volume
the ________________
of the
balloon are changing.
In each case the rate is a
derivativethat has to be
___________
computed given the rate at
which some other variable,
like time, is known to
change.
To find this derivative we write
an equation that relates the two
differentiate
variables. We then ____________
both sides of the equation with
time
respect to ________
to express
the derivative we SEEK in terms
of the derivative we KNOW.
Often the key to relating the
variables in this type of
problem is DRAWING A
PICTURE that shows the
geometric relationships
between the variables.
1.
2.
3.
4.
Identify and LABEL all the given info and
what you are asked to find. Draw a picture
if appropriate.
Write an EQUATION relating the variables.
Differentiate both sides of the equation
with respect to TIME.
Substitute and Solve. Sometimes you will
need to use the original equation or other
equations to solve for missing parts.
Water is draining from a conical tank with height 10 ft and
diameter 6 ft into a cylindrical tank that has a base with area
500π ft2. The depth h, in ft, of the water in the conical tank
is changing at a rate of (h-10) ft/min.
a) Write an expression for the radius of the water
in the conical tank, r, as a function of h.
b) Write an expression for the volume, V, of
water in the conical tank as a function of h.
c) At what rate is the volume of the water in
the conical tank changing when h=3?
d) Let y be the depth, in ft, of water in the
cylindrical tank. Write an expression for the
volume, V, of water in the cylindrical tank
as a function of y.
e) At what rate is y changing when h=3?
6 ft
h ft
10 ft
An airplane is flying on a flight path that will take it directly over a
radar tracking station. It is flying at an altitude of 6 mi, s miles from
the station. If s is decreasing at a rate of 400 mi/hr when s=10 mi,
what is the speed of the plane?
Find the rate of change in the angle of elevation of the
camera at 10 sec after lift-off.
A camera at ground level is
filming the lift-off of a
space shuttle that is rising
vertically according to the
position equation s=50t2,
where s is measured in ft
and t is measured in sec.
The camera is 2000 ft from
the launch pad.
Oil spills in a circular pattern. The radius grows at
4 ft/min. How fast is the area of oil changing
when r=10 ft.
Consider the curve defined by
x2+xy+y2=27 whose graph is
given to the right.
a) Write an expression for the
slope of the curve at any point (x,y)
b) Determine whether the lines tangent to the
curve at the x-intercepts of the curve are parallel.
Show the analysis that leads to your conclusion.
c) Find the points on the curve where the lines
tangent to the curve are vertical.
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154 #21, 22, 26, 28,
29, 31, 44, 45