the
x
-coordinate of Ship A at time
t (measured in hours after high noon) and by
the
y
-coordinate of Ship
B
, then
x
and
y
are functions of time
t.
1. A Related rates problem from Three Points of View: Pre Calculus estimate, Explicit differentiation, and Implicit differentiation.
Suppose that at noon, ship A is 150 km west of ship B . Ship A is sailing east at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4:00 p.m.?
We set up a coordinate system so that Ship
B
is at the origin at noon. Thus Ship
A
will be traveling along the
x
-axis, and Ship
B
will be traveling along the
y
-axis. If we denote by
the
x
-coordinate of Ship A at time
t (measured in hours after high noon) and by
the
y
-coordinate of Ship
B
, then
x
and
y
are functions of time
t.
Use Maple to plot the functions x and y over reasonable domains. How far apart are the two ships at 4pm? How far apart are they at 1 second after 4pm? What is the average speed of separation (in km/hr) over this one-second time interval?
Submission:
(a) A graph of x and y together on the same plot (make intelligent choices for labels and titles).
(b) A calculation of the average speed of separation over the one-second time interval.
Submission worksheet:
2. Two Calculus computations of the Instantaneous rate .
Let
s
(
t
) be the distance between the ships at time
t
. Use Maple to define this function, to plot it, and then to find
. (This is an "explicit" computation because we explicitly express the dependent variable
as a function of the independent variable
.)
Now below we
restart;
and forget that we actually know the functions
x
,
y
and
s
. We only need to remember that
s, x, y
are functions of
t
, and the basic equation relating these functions:
. We then differentiate this.
> restart;
> Equation:=s(t)^2=x(t)^2+y(t)^2;
> diff(Equation,t);
What you see is an expression with six quantities including s'(
t
). You know the other five when
t
=4, and so you can figure out s'(4)…so do this.
(This is an "implicit" computation because we do not explicitly express the dependent variable
as a function of the independent variable
.)
Submission:
(a) Your plot of
over a sensible domain, with intelligent choices for labels and title.
(b) Your work leading to the instantaneous rate of separation, s'(4), by using the explicit method and the implicit method.
(c) A summary comparing the average rate from above with the instantaneous rate. (They should be close, but not equal.)
Submission worksheet: