Section 12-6

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Sherlock Holmes followed footprints and other clues to track down suspected criminals. As he followed the clues, he knew exactly where the person had been.
The path could be drawn on a map, every location described by x- and ycoordinates.
But how could he determine when the suspect was at each place? He needed to know how x and y depended on a third variable, time.
Two variables are often not enough to describe interesting situations fully.

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You can use parametric equations to describe the x- and y-coordinates of a point separately as functions of a third variable, t, called the parameter.
Parametric equations provide you with more information and better control over which points you plot.

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Hanna’s hot-air balloon is ascending at a rate of 15 ft/s. A wind is blowing continuously from west to east at 24 ft/s. Write parametric equations to model this situation, and decide whether or not the hot-air balloon will clear power lines that are 300 ft to the east and 95 ft tall.
Find the time it takes for the balloon to touch or pass over the power lines.

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Create a table of time, ground distance, and height for a few seconds of flight.
Set the origin as the initial launching location of the balloon. Let x represent the ground distance traveled to the east in feet, and let y represent the balloon’s height above the ground in feet.
The table shows these values for the first 4 seconds of flight.

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The parametric equations that model the motion are x =24t and y =15t.
Graph this pair of equations on your calculator.

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You can picture the power lines by plotting the point (300, 95). If you trace the graph to a time of 1 s, you will see that the balloon is 24 ft to the east, at a height of 15 ft. At 12.5 s, it has traveled 300 ft to the east and has reached a height of 187.5 ft.
Hanna’s balloon will not touch the power lines.

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To solve the problem analytically, first find the time when the balloon will be 300 ft to the east. Substitute
300 for x into the equation x = 24t and solve for t: 300 = 24t, so t =12.5 s.
Then substitute this time into the equation that determines the height of Hannah’s balloon: y=15(12.5) = 187.5 ft.
These answers confirm the results you found by graphing.

Many pairs of parametric equations can be written as a single equation using only x and y.
If you can eliminate the parameter in parametric equations, then you’ll have two different ways to study a relationship.

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This investigation involves four participants: a walker, recorder X, recorder Y, and a director.
Procedural Note
1. The walker starts at one end of the segment and walks slowly for 5 s to reach the other end.
2. Recorder X points a motion sensor set for 5 s at the walker and moves along the y-axis, keeping even with the walker, thus measuring the x-coordinate of the walker’s path as a function of time.
3. Simultaneously, recorder Y points a motion sensor set for
5 s at the walker and moves along the x-axis, keeping even with the walker, thus measuring the y-coordinate of the walker’s path as a function of time.
4. The director starts all three participants at the same moment and counts out the seconds.