818
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
C
35. A hyperbola with the following graph:
y
In Problems 37–42, find the coordinates of any foci relative to
the original coordinate system.
5
(4, 4)
(⫺2, 4)
(0, 2)
37. Problem 15
(2, 2)
38. Problem 16
x
⫺5
39. Problem 17
40. Problem 18
⫺5
41. Problem 21
36. A hyperbola with the following graph:
42. Problem 22
y
5
In Problems 43–46, find the coordinates of all points of
intersection to two decimal places.
(2, 0)
(3, 1)
43. 3x2 ⫺ 5y2 ⫹ 7x ⫺ 2y ⫹ 11 ⫽ 0, 6x ⫹ 4y ⫽ 15
x
⫺5
5
44. 8x2 ⫹ 3y2 ⫺ 14x ⫹ 17y ⫺ 39 ⫽ 0, 5x ⫺ 11y ⫽ 23
(3, ⫺3)
⫺5
45. 7x2 ⫺ 8x ⫹ 5y ⫺ 25 ⫽ 0, x2 ⫹ 4y2 ⫹ 4x ⫺ y ⫺ 12 ⫽ 0
(2, ⫺2)
46. 4x2 ⫺ y2 ⫺ 24x ⫺ 2y ⫹ 35 ⫽ 0, 2x2 ⫹ 6y2 ⫺ 3x ⫺ 34 ⫽ 0
Section 10-5 Parametric Equations
Parametric Equations and Plane Curves
Parametric Equations and Conic Sections
Projectile Motion
Cycloid
Parametric Equations and Plane Curves
FIGURE 1
Graph of x ⫽ t ⫹ 1, y ⫽ t2 ⫺ 2t,
⫺⬁ ⬍ t ⬍ ⬁.
y
Consider the two equations
x⫽t⫹1
y ⫽ t 2 ⫺ 2t
10
5
x
⫺⬁ ⬍ t ⬍ ⬁
(1)
Each value of t determines a value of x, a value of y, and hence, an ordered pair
(x, y). To graph the set of ordered pairs (x, y) determined by letting t assume all
real values, we construct Table 1 listing selected values of t and the corresponding values of x and y. Then we plot the ordered pairs (x, y) and connect them
with a continuous curve, as shown in Figure 1. The variable t is called a parameter and does not appear on the graph. Equations (1) are called parametric equations because both x and y are expressed in terms of the parameter t. The graph
of the ordered pairs (x, y) is called a plane curve.
10-5 Parametric Equations
T A B L E
819
1
t
0
1
2
3
4
ⴚ1
ⴚ2
x
1
⫺2
3
4
5
0
⫺1
y
0
⫺1
0
3
8
3
⫺8
Parametric equations can also be graphed on a graphing utility. Figure 2(a)
shows the Parametric mode selected on a Texas Instruments TI-83 calculator. Figure 2(b) shows the equation editor with the parametric equations in (1) entered
as x1T and y1T. In Figure 2(c), notice that there are three new window variables,
Tmin, Tmax, and Tstep, that must be entered by the user.
FIGURE 2
Graphing parametric equations
on a graphing utility.
10
⫺3
7
⫺2
(a)
(b)
Explore/Discuss
1
(c)
(d)
(A) Consult the manual for your graphing utility and reproduce
Figure 2(a).
(B) Discuss the effect of using different values for Tmin and Tmax. Try
Tmin ⫽ ⫺1 and ⫺3. Try Tmax ⫽ 3 and 5.
(C) Discuss the effect of using different values for Tstep. Try
Tstep ⫽ 1, 0.1, and 0.01.
In some cases it is possible to eliminate the parameter by solving one of the
equations for t and substituting into the other. In the example just considered,
solving the first equation for t in terms of x, we have
t⫽x⫺1
Then, substituting the result into the second equation, we obtain
y ⫽ (x ⫺ 1)2 ⫺ 2(x ⫺ 1)
⫽ x2 ⫺ 4x ⫹ 3
We recognize this as the equation of a parabola, as we would guess from Figure 1.
820
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
In other cases, it may not be easy or possible to eliminate the parameter to
obtain an equation in just x and y. For example, for
x ⫽ t ⫹ log t
y ⫽ t ⫺ et
t⬎0
you will not find it possible to solve either equation for t in terms of functions
we have considered.
Is there more than one parametric representation for a plane curve? The answer
is yes. In fact, there is an unlimited number of parametric representations for the
same plane curve. The following are two additional representations of the parabola
in Figure 1.
x⫽t⫹3
y ⫽ t 2 ⫹ 2t
x⫽t
y ⫽ t 2 ⫺ 4t ⫹ 3
⫺⬁ ⬍ t ⬍ ⬁
(2)
⫺⬁ ⬍ t ⬍ ⬁
(3)
The concepts introduced in the preceding discussion are summarized in Definition 1.
DEFINITION
1
PARAMETRIC EQUATIONS AND PLANE CURVES
A plane curve is the set of points (x, y) determined by the parametric
equations
x ⫽ f(t)
y ⫽ g(t)
where the parameter t varies over an interval I and the functions f and g
are both defined on the interval I.
Why are we interested in parametric representations of plane curves? It turns
out that this approach is more general than using equations with two variables as
we have been doing. In addition, the approach generalizes to curves in three- and
higher-dimensional spaces. Other important reasons for using parametric representations of plane curves will be brought out in the discussion and examples that
follow.
EXAMPLE
1
Eliminating the Parameter
Eliminate the parameter and identify the plane curve given parametrically by
x ⫽ 兹t
y ⫽ 兹9 ⫺ t
0ⱕtⱕ9
(4)
10-5 Parametric Equations
Solution
821
To eliminate the parameter t, we solve each equation in (4) for t:
y ⫽ 兹9 ⫺ t
x ⫽ 兹t
x2 ⫽ t
y2 ⫽ 9 ⫺ t
t ⫽ 9 ⫺ y2
Equating the last two equations, we have
x2 ⫽ 9 ⫺ y2
x2 ⫹ y2 ⫽ 9
A circle of radius 3 centered at (0, 0)
Thus, the graph of the parametric equations in (4) is the quarter of the circle of
radius 3 centered at the origin that lies in the first quadrant (Fig. 3).
FIGURE 3
3
⫺4.5
4.5
⫺3
(a)
MATCHED PROBLEM
1
(b)
Eliminate the parameter and identify the plane curve given parametrically by
x ⫽ 兹4 ⫺ t, y ⫽ ⫺ 兹t, 0 ⱕ t ⱕ 4.
Parametric Equations and Conic Sections
Trigonometric functions provide very effective representations for many conic
sections. The following examples illustrate the basic concepts.
EXAMPLE
2
Identifying a Conic Section in Parametric Form
Eliminate the parameter ␪ and identify the plane curve given by
x ⫽ 8 cos ␪
y ⫽ 4 sin ␪
Solution
0 ⱕ ␪ ⱕ 2␲
(5)
To eliminate the parameter ␪, we solve the first equation in (5) for cos ␪, the second for sin ␪, and substitute into the Pythagorean identity cos2 ␪ ⫹ sin2 ␪ ⫽ 1:
cos ␪ ⫽
x
8
and
sin ␪ ⫽
y
4
822
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
FIGURE 4
Graph of x ⫽ 8 cos ␪, y ⫽ 4 sin ␪,
0 ⱕ ␪ ⱕ 2␲.
6
cos2 ␪ ⫹ sin2 ␪ ⫽ 1
冢 冣 冢 冣 ⫽1
x
8
2
⫹
y
4
2
y2
x2
⫹
⫽1
64 16
⫺9
9
The graph is an ellipse (Fig. 4).
⫺6
Eliminate the parameter ␪ and identify the plane curve given by x ⫽ 4 cos ␪,
y ⫽ 4 sin ␪, 0 ⱕ ␪ ⱕ 2␲.
MATCHED PROBLEM
2
Explore/Discuss
2
Graph one period (0 ⱕ ␪ ⱕ 2␲) of each of the three plane curves given
parametrically by
x1 ⫽ 5 cos ␪
x2 ⫽ 2 cos ␪
x3 ⫽ 5 cos ␪
y1 ⫽ 5 sin ␪
y2 ⫽ 2 sin ␪
y3 ⫽ 2 sin ␪
Identify the curves by eliminating the parameter. What happens if you
graph less than one period? More than one period?
EXAMPLE
Parametric Equations for Conic Sections
3
Find parametric equations for the conic section with the given equation:
(A) 25x2 ⫹ 9y2 ⫺ 100x ⫹ 54y ⫺ 44 ⫽ 0
(B) x2 ⫺ 16y2 ⫺ 10x ⫹ 32y ⫺ 7 ⫽ 0
(A) By completing the square in x and y we obtain the standard form
(x ⫺ 2)2 (y ⫹ 3)2
⫹
⫽ 1. So the graph is an ellipse with center (2, ⫺3) and
9
25
major axis on the line x ⫽ 2. Since cos2 ␪ ⫹ sin2 ␪ ⫽ 1, a parametric
x⫺2
representation with parameter ␪ is obtained by letting
⫽ cos ␪,
3
y⫹3
⫽ sin ␪:
5
Solutions
FIGURE 5
x ⫽ 2 ⫹ 3 cos ␪, y ⫽ ⫺3 ⫹
5 sin ␪, 0 ⱕ ␪ ⱕ 2␲.
3
⫺9
9
x ⫽ 2 ⫹ 3 cos ␪
y ⫽ ⫺3 ⫹ 5 sin ␪
⫺9
Since sin ␪ and cos ␪ have period 2␲, graphing these equations for 0 ⱕ ␪
ⱕ 2␲ will produce a complete graph of the ellipse (Fig. 5).
10-5 Parametric Equations
(B) By completing the square in x and y we obtain the standard form
(x ⫺ 5)2
⫺ (y ⫺ 1)2 ⫽ 1. So the graph is a hyperbola with center (5, 1) and
16
transverse axis on the line y ⫽ 1. Since sec2 ␪ ⫺ tan2 ␪ ⫽ 1, a parametric
x⫺5
representation with parameter ␪ is obtained by letting
⫽ sec ␪,
4
y ⫺ 1 ⫽ tan ␪:
FIGURE 6
x ⫽ 5 ⫹ 4 sec ␪, y ⫽ 1 ⫹ tan ␪,
␲ 3␲
0 ⱕ ␪ ⱕ 2␲, ␪ ⫽ , .
2 2
6
⫺2.5
823
12.5
x ⫽ 5 ⫹ 4 sec ␪
y ⫽ 1 ⫹ tan ␪
⫺4
The period of tan ␪ is ␲, but the period of sec ␪ is 2␲, so we have to use
0 ⱕ ␪ ⱕ 2␲ to produce a complete graph of the hyperbola (Fig. 6). To be
precise, we should exclude ␪ ⫽ ␲/2 and 3␲/2, since the tangent function is
not defined at these values. Including them does not affect the graph, since
most graphing utilities ignore points where functions are undefined. Note that
when the parametric equations are graphed in the connected mode, the graph
appears to show the asymptotes of the hyperbola (see Fig. 6).
MATCHED PROBLEM
3
Remark
Find parametric equations for the conic section with the given equation.
(A) 36x2 ⫹ 16y2 ⫹ 504x ⫺ 96y ⫹ 1,332 ⫽ 0
(B) 16y2 ⫺ 9x2 ⫺ 36x ⫹ 128y ⫹ 76 ⫽ 0
Refer to Example 3, part A. Any interval of the form a ⱕ ␪ ⱕ a ⫹ b, where
b ⱖ 2␲, will produce a graph containing all the points on this ellipse. We will
follow the practice of always choosing the shortest interval starting at 0 that will
generate all the points on a conic section. For this ellipse, that interval is [0, 2␲].
Projectile Motion
Newton’s laws and advanced mathematics can be used to determine the path of
a projectile. If v0 is the initial speed of the projectile at an angle ␣ with the horizontal and a0 is the initial altitude of the projectile (see Fig. 7), then, neglecting
air resistance, the path of the projectile is given by
x ⫽ (v0 cos ␣)t
y ⫽ a0 ⫹ (v0 sin ␣)t ⫺ 4.9t 2
0ⱕtⱕb
(6)
The parameter t represents time in seconds, and x and y are distances measured
in meters. Solving the first equation in equations (6) for t in terms of x, substituting into the second equation, and simplifying, produces the following equation:
y ⫽ a0 ⫹ (tan ␣)x ⫺
4.9
x2
v cos2 ␣
2
0
You should verify this by supplying the omitted details.
(7)
824
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
FIGURE 7
y
Projectile motion.
v0
␣
v 0 cos ␣
a0
v 0 sin ␣
x
We recognize equation (7) as a parabola. This equation in x and y describes
the path the projectile follows but tells us little else about its flight. On the other
hand, the parametric equations (6) not only determine the path of the projectile
but also tell us where it is at any time t. Furthermore, using concepts from physics
and calculus, the parametric equations can be used to determine the velocity and
acceleration of the projectile at any time t. This illustrates another advantage of
using parametric representations of plane curves.
EXAMPLE
4
Projectile Motion
An automobile drives off a 50-meter cliff traveling at 25 meters per second
(see Fig. 8). When (to the nearest tenth of a second) will the automobile strike
the ground? How far (to the nearest meter) from the base of the cliff is the
point of impact?
FIGURE 8
50 m
Solution
At the instant the automobile leaves the cliff, the velocity is 25 meters per second, the angle with the horizontal is 0, and the altitude is 50 meters. Substituting
these values in equations (6), the parametric equations for the path of the automobile are
x ⫽ 25t
y ⫽ 50 ⫺ 4.9t2
The automobile strikes the ground when y ⫽ 0. Using the parametric equation
for y, we have
y ⫽ 50 ⫺ 4.9t 2 ⫽ 0
⫺4.9t 2 ⫽ ⫺50
t⫽
兹
⫺50
⬇ 3.2 seconds
⫺4.9
10-5 Parametric Equations
825
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
4
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?
The range of a projectile at an altitude a0 ⫽ 0 is the distance from the point
of firing to the point of impact. If we keep the initial speed v0 of the projectile
constant and vary the angle ␣ in Figure 7, we obtain different parabolic paths followed by the projectile and different ranges. The maximum range is obtained
when ␣ ⫽ 45°. Furthermore, assuming that the projectile always stays in the same
vertical plane, then there are points in the air and on the ground that the projectile cannot reach, irrespective of the angle ␣ used, 0° ⱕ ␣ ⱕ 180°. Using more
advanced mathematics, it can be shown that the reachable region is separated from
the nonreachable region by a parabola called an envelope of the other parabolas
(see Fig. 9).
FIGURE 9
Reachable region of a projectile.
␣
Envelope
Cycloid
We now consider an unusual curve called a cycloid, which has a fairly simple
parametric representation and a very complicated representation in terms of x and
y only. The path traced by a point on the rim of a circle that rolls along a line is
called a cycloid. To derive parametric equations for a cycloid we roll a circle of
radius a along the x axis with the tracing point P on the rim starting at the origin (see Fig. 10).
FIGURE 10
y
Cycloid.
P (x, y)
a
␪
O R
C
Q
S
x
826
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Since the circle rolls along the x axis without slipping (refer to Fig. 10), we
see that
d(O, S) ⫽ arc PS
⫽ a␪
␪ in radians
(8)
where S is the point of contact between the circle and the x axis. Referring to triangle CPQ, we see that
d(P, Q) ⫽ a sin ␪
0 ⱕ ␪ ⱕ ␲/2
(9)
d(Q, C) ⫽ a cos ␪
0 ⱕ ␪ ⱕ ␲/2
(10)
Using these results, we have
x ⫽ d(O, R)
⫽ d(O, S) ⫺ d(R, S)
⫽ (arc PS) ⫺ d(P, Q)
⫽ a␪ ⫺ a sin ␪
Use equations (8) and (9).
y ⫽ d(R, P)
⫽ d(S, C) ⫺ d(Q, C)
⫽ a ⫺ a cos ␪
Use equation (10) and the fact that
d(S, C ) ⫽ a.
Even though ␪ in equations (9) and (10) was restricted so that 0 ⱕ ␪ ⱕ ␲/2, it
can be shown that the derived parametric equations generate the whole cycloid
for ⫺⬁ ⬍ ␪ ⬍ ⬁. The graph specifies a periodic function with period 2␲a. Thus,
in general, we have Theorem 1.
THEOREM
1
PARAMETRIC EQUATIONS FOR A CYCLOID
For a circle of radius a rolled along the x axis, the resulting cycloid generated by a point on the rim starting at the origin is given by
x ⫽ a␪ ⫺ a sin ␪
y ⫽ a ⫺ a cos ␪
⫺⬁ ⬍ ␪ ⬍ ⬁
FIGURE 11
Cycloid path.
P
Q
The cycloid is a good example of a curve that is very difficult to represent
without the use of a parameter. A cycloid has a very interesting physical property. An object sliding without friction from a point P to a point Q lower than P,
but not on the same vertical line as P, will arrive at Q in a shorter time traveling along a cycloid than on any other path (see Fig. 11).
10-5 Parametric Equations
Explore/Discuss
3
827
(A) Let Q be a point b units from the center of a wheel of radius a,
where 0 ⬍ b ⬍ a. If the wheel rolls along the x axis with the tracing point Q starting at (0, a ⫺ b), explain why parametric equations
for the path of Q are given by
x ⫽ a ␪ ⫺ b sin ␪
y ⫽ a ⫺ b cos ␪
(B) Use a graphing utility to graph the paths of a point on the rim of a
wheel of radius 1, and a point halfway between the rim and center,
as the wheel makes two complete revolutions rolling along the x
axis.
Answers to Matched Problems
1. The quarter of the circle of radius 2 centered at the origin that lies in the fourth quadrant.
2. x2 ⫹ y2 ⫽ 16, circle of radius 4 centered at (0, 0)
3. (A) Ellipse: x ⫽ ⫺7 ⫹ 4 cos ␪, y ⫽ 3 ⫹ 6 sin ␪, 0 ⱕ ␪ ⱕ 2␲
␲ 3␲
(B) Hyperbola: x ⫽ ⫺2 ⫹ 4 tan ␪, y ⫽ ⫺4 ⫹ 3 sec ␪, 0 ⱕ ␪ ⱕ 2␲, ␪ ⫽ ,
2 2
4. 2.8 meters
EXERCISE 10-5
B
A
In Problems 13–24, obtain an equation in x and y by eliminating the parameter. Identify the curve.
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.
13. x ⫽ t ⫺ 2, y ⫽ 4 ⫺ 2t
2. If x ⫽ t 2 and y ⫽ t 4 ⫺ 2, then y ⫽ x2 ⫺ 2. Discuss the differences between the graph of the parametric equations
and the graph of the parabola y ⫽ x2 ⫺ 2.
15. x ⫽ t ⫺ 1, y ⫽ 兹t, t ⱖ 0
14. x ⫽ t ⫺ 1, y ⫽ 2t ⫹ 2
16. x ⫽ 兹t, y ⫽ t ⫹ 1, t ⱖ 0
17. x ⫽ 兹t, y ⫽ 2兹16 ⫺ t, 0 ⱕ t ⱕ 16
18. x ⫽ ⫺3兹t, y ⫽ 兹25 ⫺ t, 0 ⱕ t ⱕ 25
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 ⫽ 14t 2, y ⫽ t
10. x ⫽ 2t, y ⫽ t 2
11. x ⫽ 14t 4, y ⫽ t2
12. x ⫽ 2t2, y ⫽ t4
19. x ⫽ ⫺兹t ⫹ 1, y ⫽ ⫺兹t ⫺ 1, t ⱖ 1
20. x ⫽ 兹2 ⫺ t, y ⫽ ⫺兹4 ⫺ t, t ⱕ 2
21. x ⫽ 3 sin ␪, y ⫽ 4 cos ␪, 0 ⱕ ␪ ⱕ 2␲
22. x ⫽ 3 sin ␪, y ⫽ 3 cos ␪, 0 ⱕ ␪ ⱕ 2␲
23. x ⫽ 2 ⫹ 2 sin ␪, y ⫽ 3 ⫹ 2 cos ␪, 0 ⱕ ␪ ⱕ 2␲
24. x ⫽ 3 ⫹ 4 sin ␪, y ⫽ 2 ⫹ 2 cos ␪, 0 ⱕ ␪ ⱕ 2␲
25. If A ⫽ 0, C ⫽ 0, and E ⫽ 0, find parametric equations for
Ax2 ⫹ Cy2 ⫹ Dx ⫹ Ey ⫹ F ⫽ 0. Identify the curve.
828
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
26. If A ⫽ 0, C ⫽ 0, and D ⫽ 0, find parametric equations for
Ax2 ⫹ Cy2 ⫹ Dx ⫹ Ey ⫹ F ⫽ 0. Identify the curve.
In Problems 27–30, eliminate the parameter and find the
standard equation for the curve. Name the curve and find its
center.
27. x ⫽ 3 ⫹ 6 cos t, y ⫽ 2 ⫹ 4 sin t, 0 ⱕ t ⱕ 2␲
␲ 3␲
28. x ⫽ 1 ⫹ 3 sec t, y ⫽ ⫺2 ⫹ 2 tan t, 0 ⱕ t ⱕ 2␲, t ⫽ ,
2 2
29. x ⫽ ⫺3 ⫹ 2 tan t, y ⫽ ⫺1 ⫹ 5 sec t, 0 ⱕ t ⱕ 2␲,
␲ 3␲
t⫽ ,
2 2
41. Consider the following two pairs of parametric equations:
1. x1 ⫽ t, y1 ⫽ et, ⫺⬁ ⬍ t ⬍ ⬁
2. x2 ⫽ et, y2 ⫽ t, ⫺⬁ ⬍ t ⬍ ⬁
(A) Graph both pairs of parametric equations in a squared
viewing window and discuss the relationship between
the graphs.
(B) Eliminate the parameter and express each equation as
a function of x. How are these functions related?
42. Consider the following two pairs of parametric equations:
1. x1 ⫽ t, y1 ⫽ log t, t ⬎ 0
2. x2 ⫽ log t, y2 ⫽ t, t ⬎ 0
30. x ⫽ ⫺4 ⫹ 5 cos t, y ⫽ 1 ⫹ 8 sin t, 0 ⱕ t ⱕ 2␲
(A) Graph both pairs of parametric equations in a squared
viewing window and discuss the relationship between
the graphs.
C
(B) Eliminate the parameter and express each equation as
a function of x. How are these functions related?
In Problems 31–36, the interval for the parameter is the entire
real line. Obtain an equation in x and y by eliminating the
parameter and identify the curve.
31. x ⫽ 兹t 2 ⫹ 1, y ⫽ 兹t 2 ⫹ 9
32. x ⫽ 兹t 2 ⫹ 4, y ⫽ 兹t 2 ⫹ 1
33. x ⫽
2
2t
,y⫽
兹t 2 ⫹ 1
兹t 2 ⫹ 1
34. x ⫽
3
3t
,y⫽
2
兹t ⫹ 1
兹t ⫹ 1
35. x ⫽
4t
8
,y⫽ 2
t2 ⫹ 4
t ⫹4
36. x ⫽
4t 2
4t
,
y
⫽
t2 ⫹ 1
t2 ⫹ 1
2
APPLICATIONS
43. Projectile Motion. An airplane flying at an altitude of
1,000 meters is dropping medical supplies to hurricane
victims on an island. The path of the plane is horizontal,
the speed is 125 meters per second, and the supplies are
dropped at the instant the plane crosses the shoreline. How
far inland (to the nearest meter) will the supplies land?
44. Projectile Motion. One stone is dropped vertically from
the top of a tower 40 meters high. A second stone is
thrown horizontally from the top of the tower with a speed
of 30 meters per second. How far apart (to the nearest
tenth of a meter) are the stones when they land?
45. Projectile Motion. A projectile is fired with an initial
speed of 300 meters per second at an angle of 45° to the
horizontal. Neglecting air resistance, find
In Problems 37–40, find the standard form of each equation.
Name the curve and find its center. Then use trigonometric
functions to find parametric equations for the curve.
(A) The time of impact
37. 25x2 ⫺ 200x ⫺ 9y2 ⫺ 18y ⫹ 616 ⫽ 0
(C) The maximum height in meters of the projectile
38. 36x ⫹ 360x ⫹ 4y ⫺ 8y ⫹ 760 ⫽ 0
Compute all answers to three decimal places.
2
2
39. 4x2 ⫺ 24x ⫹ 49y2 ⫹ 392y ⫹ 624 ⫽ 0
40. 16x2 ⫹ 32x ⫺ 9y2 ⫺ 36y ⫺ 164 ⫽ 0
(B) The horizontal distance covered (range) in meters and
kilometers at time of impact
46. Projectile Motion. Repeat Problem 45 if the same projectile is fired at 40° to the horizontal instead of 45°.