Parametric Equations
Two ships, the Andor and the Helsinki, are sailing in the fog. The same
tracking equipment is monitoring both ships. As the ships enter the
area being monitored, their positions are displayed on the observer’s
rectangular screen by a brightened point. The point representing the
Andor is at a point 900 mm from the bottom left corner of the screen
along the lower edge of the screen. The point representing the Helsinki
is located 100mm directly above the lower left hand corner. One minute
later the positions have changed. The Andor has moved to a location on
the screen that is 3 mm west and 2 mm north of its previous location.
The Helsinki has moved to a position 4 mm east and 1 mm north of
where it had been. Assume that the two ships will continue to sail at a
constant speed on their respective linear paths.
On the grid below, you will see the points representing the position of
the two ships as the first appear on the tracking screen. On this grid,
label each point with an H for Helsinki or an A for Andor and sketch the
linear paths of each ship. Then, state whether the two ships, if they
continue on their respective paths, will collide(crash). Defend your
answer mathematically.
Graph the parametric equations.
Show algebraically that the parametric
equations for the Andor are
equivalent to the linear function that you
found.
Explain what the coefficient of the variables and the
constant mean in the function
x(t) = at + b, y(t) = ct + d
and how they relate to the linear function y(x).
Your Turn
Plane A is spotted on a rectangular tracking screen, by an air traffic
controller at (420,0) flying west 2 miles and north 3.2 miles each
minute.
1. Write a linear equation to represent the path of the plane.
2. What is the speed of the plane? (Be sure to include units)
3. Write the parametric equations that model the location of
the plane, t minutes after it has entered the screen.
Plane B enters the screen and its path is given by the parametric
equations x = t and y = 300 + 1.5t. The planes are flying at the
same altitude.
4. Find the point at which the paths of plane A and plane B
intersect.
5. Find the amount of time it takes Plane A and Plane B to
get to the point of intersection. Do the planes crash?
Sketching a graph
Sketch the curve
represented by the
parametric equations
(indicate the direction
using arrows) for -3 < t < 3.
2
x  t 3
y  2t
Eliminate the parameter and
write the corresponding
rectangular equation whose
graph represents the curve.
Parametric Equations Investigation
Along the River Thames in London, England, there is a huge
Ferris wheel known as The British Airways London Eye, but often
called the Millennium Ferris Wheel. The Wheel is 135 m (just
under 450 ft) high and makes one complete revolution every 15
minutes, in a counterclockwise direction, when the ride is
underway. For this exploration, we will reduce the immense
passenger capsules to mere points on the rim of a circular wheel.
The capsules must clear the ground in their rotation, but we will
assume that the radius of the Wheel is the full 67.5 m —
exactly one-half the height of the Wheel. This will allow our
model to reflect ground level access to the capsule for its
passengers.
Recall the radius of the London Eye Ferris wheel
was 67.5 meters. Write parametric equations to
represent a circle with radius 67.5 meters.
Also recall that the wheel revolves once every 15
minutes. How far will the wheel rotate in 1 minute?
This is the angular velocity and is the coefficient of t in our parametric
equations.
3) Let the point (0, 0) correspond to the location of a
capsule when it is at ground level and let t = 0 represent
the time when a particular capsule is at ground level.
Write parametric equations that represent the motion of
this capsule on the London Eye over the course of a 15
minute ride.
Graph the equations on your calculator in an appropriate
viewing window. What are the coordinates of the capsule
at the beginning and end of the ride?
How high off the ground is the capsule after 10 minutes?