Chapter Projects 797

1. The Orbits of Neptune and Pluto The orbit ofa planet about the Sun is an ellipse, with the Sun at onefocus.The aphelion of a planet is its greatest distance fromthe Sun and the perihelion is its shortest distance. Themean distance of a planet from the Sun is the length ofthe semimajor axis of the elliptical orbit. See theillustration.

(a) The aphelion of Neptune is and itsperihelion is Find the equation forthe orbit of Neptune around the Sun.

(b) The aphelion of Pluto is and itsperihelion is Find the equation forthe orbit of Pluto around the Sun.

(c) Graph the orbits of Pluto and Neptune on a graphingutility. Notice that the graphs of the orbits of theplanets do not intersect! But, in fact, the orbits dointersect. What is the explanation?

(d) The graphs of the orbits have the same center, so theirfoci lie in different locations. To see an accurate rep-resentation, the location of the Sun (a focus) needs tobe the same for both graphs.This can be accomplishedby shifting Pluto’s orbit to the left. The shift amountis equal to Pluto’s distance from the center [in thegraph in part (c)] to the Sun minus Neptune’s distancefrom the center to the Sun. Find the new equationrepresenting the orbit of Pluto.

(e) Graph the equation for the orbit of Pluto found inpart (d) along with the equation of the orbit ofNeptune. Do you see that Pluto’s orbit is sometimesinside Neptune’s?

(f) Find the point(s) of intersection of the two orbits.(g) Do you think two planets ever collide?

4445.8 * 106 km.7381.2 * 106 km

4458.0 * 106 km.4532.2 * 106 km

Majoraxis

SunCenter

PerihelionAphelion

Mean distance

2. Constructing a Bridge over the East RiverA new bridge is to be constructed over the East River inNew York City.The space between the supports needs tobe 1050 feet; the height at the center of the arch needs tobe 350 feet. Two structural possibilities exist: the supportcould be in the shape of a parabola or the support couldbe in the shape of a semiellipse.

An empty tanker needs a 280-foot clearance to passbeneath the bridge. The width of the channel for each ofthe two plans must be determined to verify that the tankercan pass through the bridge.(a) Determine the equation of a parabola with these

characteristics.

[Hint: Place the vertex of the parabola at the originto simplify calculations.]

(b) How wide is the channel that the tanker can passthrough?

(c) Determine the equation of a semiellipse with thesecharacteristics.

[Hint: Place the center of the semiellipse at the originto simplify calculations.]

(d) How wide is the channel that the tanker can passthrough?

(e) If the river were to flood and rise 10 feet, how wouldthe clearances of the two bridges be affected? Doesthis affect your decision as to which design to choose?Why?

3. Systems of Parametric Equations Considerthe following systems of parametric equations:

I.

II.

III.x2 = 2 cos t, y2 = 4 sin t, 0 … t … 2px1 = 3 sin t, y1 = 4 cos t + 2, 0 … t … 2p

x2 = t3>2, y2 = 2t + 4, t Ú 0

x1 = ln t, y2 = t3, t 7 0

x2 = sec2 t, y2 = tan2 t, 0 … t …p

4

x1 = 4t - 2, y1 = 1 - t, - q 6 t 6 q

ChannelWidth

350'

280'350'Channel

Width

280'

280'

280'

1050'

1050'

Chapter Projects

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798 CHAPTER 11 Analytic Geometry

(a) For system I, set and and solve eachequation for t. If you can solve the resulting equationsalgebraically, do so. If they cannot be solved alge-braically, solve them graphically, using your graphingcalculator. Remember that the value of t must be thesame for both the x and y equations in order to havea solution for the system.

(b) Now graph the system of parametric equations usingyour graphing calculator and find the point(s) ofintersection, if there are any. (You will need to use theTRACE feature to do this. Make sure that the samevalue of t gives any points of intersection of eachcurve.) What did you notice?

(c) Did any solutions you found in part (b) match any of

y1 = y2x1 = x2 those that you found in part (a)? Why or why not?Explain.

(d) Convert the parametric equations in system I torectangular coordinates and state the domain andrange for each equation. Find the solution to thesystem either algebraically or graphically. How doesthis solution compare to what you found in part (c)?

(e) Repeat parts (a)–(d) for system II.(f) Repeat parts (a)–(d) for system III.(g) Which method is more efficient—solving in the

parametric form or solving in rectangular form? Doesthis depend on the equations? What must you watchfor when solving systems of parametric equations?Explain.

1. Find all the solutions of the equation 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

and radius 4. Graph this circle.

4. What is the domain of the function

5. For find

6. (a) Find the domain and range of (b) Find the inverse of and state its domain

and range.7. Solve the equation 8. For what numbers x is 9. Solve the equation where

10. Find an equation for each of the following graphs:(a) Line:

(b) Circle:

(c) Ellipse:

x

y

2

3–3

–2

x

y2

2–1 4

x

y2

1

–2

0° 6 u 6 90°.cot12u2 = 1,6 - x Ú x2

?9x4 + 33x3 - 71x2 - 57x - 10 = 0.

y = 3x + 2y = 3x + 2.

f1x + h2 - f1x2h

, h Z 0.f1x2 = -3x2 + 5x - 2,

f1x2 =3

sin x + cos x ?

10, 42

sin12u2 = 0.5. (d) Parabola:

(e) Hyperbola:

(f) Exponential:

11. If (a) Solve (b) Solve f1x2 … 2.

f1x2 = 2.f1x2 = log41x - 22:

(1, 4)

x

y

(0, 1) (�1, ) 1–4

x

y

2

2–2

–2

(3, 2)

x

y

2

1–1

Cumulative Review

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