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SULL-PRECU-11-721-798.I
1/29/03
10:16 AM
Page 797
Chapter Projects
797
Chapter Projects
2.
Constructing a Bridge over the East River
A new bridge is to be constructed over the East River in
New York City. The space between the supports needs to
be 1050 feet; the height at the center of the arch needs to
be 350 feet. Two structural possibilities exist: the support
could be in the shape of a parabola or the support could
be in the shape of a semiellipse.
An empty tanker needs a 280-foot clearance to pass
beneath the bridge. The width of the channel for each of
the two plans must be determined to verify that the tanker
can pass through the bridge.
(a) Determine the equation of a parabola with these
characteristics.
1.
[Hint: Place the vertex of the parabola at the origin
to simplify calculations.]
(b) How wide is the channel that the tanker can pass
through?
(c) Determine the equation of a semiellipse with these
characteristics.
The Orbits of Neptune and Pluto The orbit of
a planet about the Sun is an ellipse, with the Sun at one
focus.The aphelion of a planet is its greatest distance from
the Sun and the perihelion is its shortest distance. The
mean distance of a planet from the Sun is the length of
the semimajor axis of the elliptical orbit. See the
illustration.
[Hint: Place the center of the semiellipse at the origin
to simplify calculations.]
(d) How wide is the channel that the tanker can pass
through?
(e) If the river were to flood and rise 10 feet, how would
the clearances of the two bridges be affected? Does
this affect your decision as to which design to choose?
Why?
Mean distance
Aphelion
Center
Perihelion
Major
axis
Sun
1050'
1050'
3.
350'
280'
280'
l
ne
an h
Ch idt
W
280'
l
ne
an h
Ch idt
W
(a) The aphelion of Neptune is 4532.2 * 106 km and its
perihelion is 4458.0 * 106 km. Find the equation for
the orbit of Neptune around the Sun.
(b) The aphelion of Pluto is 7381.2 * 106 km and its
perihelion is 4445.8 * 106 km. Find the equation for
the orbit of Pluto around the Sun.
(c) Graph the orbits of Pluto and Neptune on a graphing
utility. Notice that the graphs of the orbits of the
planets do not intersect! But, in fact, the orbits do
intersect. What is the explanation?
(d) The graphs of the orbits have the same center, so their
foci lie in different locations. To see an accurate representation, the location of the Sun (a focus) needs to
be the same for both graphs.This can be accomplished
by shifting Pluto’s orbit to the left. The shift amount
is equal to Pluto’s distance from the center [in the
graph in part (c)] to the Sun minus Neptune’s distance
from the center to the Sun. Find the new equation
representing the orbit of Pluto.
(e) Graph the equation for the orbit of Pluto found in
part (d) along with the equation of the orbit of
Neptune. Do you see that Pluto’s orbit is sometimes
inside Neptune’s?
(f) Find the point(s) of intersection of the two orbits.
(g) Do you think two planets ever collide?
350'
280'
Systems of Parametric Equations Consider
the following systems of parametric equations:
I.
II.
x1 = 4t - 2, y1 = 1 - t,
-q 6 t 6 q
p
x2 = sec2 t, y2 = tan2 t, 0 … t …
4
3
x1 = ln t, y2 = t , t 7 0
x2 = t3>2,
y2 = 2t + 4, t Ú 0
III. x1 = 3 sin t, y1 = 4 cos t + 2, 0 … t … 2p
x2 = 2 cos t, y2 = 4 sin t, 0 … t … 2p
SULL-PRECU-11-721-798.I
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CHAPTER 11
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Analytic Geometry
(a) For system I, set x1 = x2 and y1 = y2 and solve each
equation for t. If you can solve the resulting equations
algebraically, do so. If they cannot be solved algebraically, solve them graphically, using your graphing
calculator. Remember that the value of t must be the
same for both the x and y equations in order to have
a solution for the system.
(b) Now graph the system of parametric equations using
your graphing calculator and find the point(s) of
intersection, if there are any. (You will need to use the
TRACE feature to do this. Make sure that the same
value of t gives any points of intersection of each
curve.) What did you notice?
(c) Did any solutions you found in part (b) match any of
(d)
(e)
(f)
(g)
those that you found in part (a)? Why or why not?
Explain.
Convert the parametric equations in system I to
rectangular coordinates and state the domain and
range for each equation. Find the solution to the
system either algebraically or graphically. How does
this solution compare to what you found in part (c)?
Repeat parts (a)–(d) for system II.
Repeat parts (a)–(d) for system III.
Which method is more efficient—solving in the
parametric form or solving in rectangular form? Does
this depend on the equations? What must you watch
for when solving systems of parametric equations?
Explain.
Cumulative Review
1. Find all the solutions of the equation sin12u2 = 0.5.
2. Find a polar equation for the line containing the origin
that makes an angle of 30° with the positive x-axis.
3. Find a polar equation for the circle with center at the point
10, 42 and radius 4. Graph this circle.
3
4. What is the domain of the function f1x2 =
?
sin x + cos x
f1x + h2 - f1x2
5. For f1x2 = -3x2 + 5x - 2, find
, h Z 0.
h
x
6. (a) Find the domain and range of y = 3 + 2.
(b) Find the inverse of y = 3x + 2 and state its domain
and range.
7. Solve the equation 9x4 + 33x3 - 71x2 - 57x - 10 = 0.
8. For what numbers x is 6 - x Ú x2?
9. Solve the equation cot12u2 = 1, where 0° 6 u 6 90°.
10. Find an equation for each of the following graphs:
(a) Line:
y
(d) Parabola:
y
2
–1
x
1
(e) Hyperbola:
y
(3, 2)
2
–2
x
2
–2
(f) Exponential:
y
2
(1, 4)
x
1
(1,
–2
1–
4)
(0, 1)
x
(b) Circle:
y
2
–1
(c) Ellipse:
11. If f1x2 = log41x - 22:
(a) Solve f1x2 = 2.
(b) Solve f1x2 … 2.
4 x
2
y
2
–3
3
–2
x