College Algebra 9th txtbk

SECTION B–3 Parametric Equations A-27
SOLUTION
At the instant the automobile leaves the cliff, the vertical speed is 0, the horizontal speed is
25 meters per second, and the altitude is 50 meters. Substituting these values in equations
(5), the parametric equations for the path of the automobile are
x 25t
The automobile strikes the ground when y 0. Using the parametric equation for y, we
have
y 50 4.9t 2 0
y 50 4.9t 2
4.9t 2 50
50
t 3.2 seconds
B 4.9
The distance from the base of the cliff is the same as the value of x. Substituting t 3.2
in the first parametric equation, the distance from the base of the cliff at the point of impact
is x 25(3.2) 80 meters.
MATCHED PROBLEM 2
A gardener is holding a hose in a horizontal position 1.5 meters above the ground. Water is
leaving the hose at a speed of 5 meters per second. What is the distance (to the nearest tenth
of a meter) from the gardener’s feet to the point where the water hits the ground?
ANSWERS TO MATCHED PROBLEMS
1. The quarter of the circle of radius 2 centered at the origin that lies in the fourth quadrant.
2. 2.8 meters
B-3 Exercises
1. If x t 2 and y t 2 2, then y x 2. Discuss the differences
between the graph of the parametric equations and the graph of
the line y x 2.
2. If x t 2 and y t 4 2, then y x 2 2. Discuss the differences
between the graph of the parametric equations and the
graph of the parabola y x 2 2.
In Problems 3–12, the interval for the parameter is the whole real
line. For each pair of parametric equations, eliminate the
parameter t and find an equation for the curve in terms of x and y.
Identify and graph the curve.
3. x t, y 2t 2 4. x t, y t 1
5. x t 2 , y 2t 2 2 6. x t 2 , y t 2 1
7. x 3t, y 2t 8. x 2t, y t
9. x 1 4t 2 , y t
10. x 2t, y t 2
11. x 1 4t
4 , y t 2
12. x 2t 2 , y t 4
In Problems 13–20, obtain an equation in x and y by eliminating
the parameter. Identify the curve.
13. x t 2, y 4 2t
14. x t 1, y 2t 2
15. x t 1, y 1t, t 0
16. x 1t, y t 1, t 0
17. x 1t, y 2116 t, 0 t 16
18. x 31t, y 125 t, 0 t 25
19. x 1t 1, y 1t 1, t 1
20. x 12 t, y 14 t, t 2
21. If A ≠ 0, C 0, and E ≠ 0, find parametric equations for
Ax 2 Cy 2 Dx Ey F 0. Identify the curve.
22. If A 0, C ≠ 0, and D ≠ 0, find parametric equations for
Ax 2 Cy 2 Dx Ey F 0. Identify the curve.