Figure 4-15 shows log-log coordinatesfor defining points in a portion of the xy- plane. Both axes are logarithmic. The numerical values that can be depicted on either axis are restricted to one sign or the other (positive or negative). In this example, the graphable ranges of the variables are:
0.1≤x≤10 0.1≤y≤10
Both of the axes in Fig. 4-15 span two orders of magnitude (powers of 10). The span of either axis could be larger or smaller, but in any case the values can- not extend to zero.
PROBLEM 4-9
How do the celestial latitude and longitude of a star in the sky, such as Sirius (the “Dog Star”), change over the course of a 24-hour period?
SOLUTION 4-9
The celestial latitude of a star does not change over the course of a 24- hour period. The celestial longitude progresses from east to west, making
x y 10 1 3 0.3 0.1 10 1 0.3 3 0.1
a complete circle every 24 hours. For example, if Sirius is at celestial longitudef =0º at 1:00 A.M. Eastern Standard Time (EST), then f = –90º at 7:00 A.M. EST, f= +180º at 1:00 P.M. EST, and f= +90º at 7:00 P.M. EST. In the case of a star that is exactly at the north celestial pole or the south celestial pole, the celestial longitude is not defined. Polaris (the “North Star”) is almost exactly at the north celestial pole.
PROBLEM 4-10
How do the declination and right ascension of a star in the sky, such as Sirius (the “Dog Star”), change over the course of a 24-hour period?
SOLUTION 4-10
Neither of these coordinates changes over the course of a 24-hour period, because declination and right ascension are defined with respect to the stars, not with respect to the surface of the earth. This is why astronomers prefer to specify declination and right ascension, rather than celestial latitude and longitude, for locating stars, nebulae, galaxies, and other distant objects in the heavens.
Quick Practice
Here are some practice problems that cover the material presented in this chap- ter. Solutions follow the problems.
PROBLEMS
1. Find the distance between the points (2,4) and (–3,12) on the Cartesian plane. Assume these coordinates are exact, and round the answer off to three significant figures.
2. Find the distance between the points (0,0) and (100,110) on the Cartesian plane. Assume these coordinates are exact, and round the answer off to five significant figures. Use scientific notation for all numerical values of 10,000 (104) or greater.
3. Suppose a line has a slope of 3 and passes through the point (0,–8) on the Cartesian plane. What is the equation of this line in slope-intercept form?
4. Suppose a line is parallel to the yaxis, and runs through the point (2,0) on the Cartesian plane. What is the equation of this line? What is the slope of this line?
5. Suppose a line runs through the point (–4,7) and has a slope of –2 on the Cartesian plane. What is the equation of this line in slope-intercept form? In standard form?
SOLUTIONS
1. Let (x0,y0)= (2,4), and let (x1,y1)=(–3,12). The distance dbetween the points is: d=[(x1–x0)2+(y 1–y0) 2]1/2 =[(–3 – 2)2+(12 – 4)2)]1/2 =[(–5)2+82]1/2 =(25+64)1/2 =891/2=9.433981. . .
This rounds off to 9.43.
2. Let (x0,y0)=(0,0), and let (x1,y1)=(100,110). The distance dbetween the points is: d=[(x1–x0)2+(y 1–y0) 2]1/2 =[(100 – 0)2+(110 – 0)2]1/2 =(1002+1102)1/2 =(1.0000×104+1.2100×104)1/2 =(2.2100×104)1/2 =148.660687. . . This rounds off to 148.66.
3. The slope-intercept form for a line is y=mx+k, where mis the slope and kis the y-intercept. We are given m=3. The point (0,–8) is on the yaxis becausex=0, so the y-intercept is –8. The equation is found by plugging in these numbers, as follows:
y=3x +(–8) y=3x– 8
4. The equation of the line is simply x=2. The slope is undefined, because the line is “vertical” (parallel to the yaxis).
5. The point-slope form for a line is y–y0=m(x–x0), where mis the slope and (x0,y0) is a point on the line. We are given (x0,y0)= (–4,7) and m = –2. The equation in slope-intercept form is found by “plugging numbers in” and then simplifying, as follows:
y–y0=m(x–x0) y– 7 =–2[x– (–4)] y– 7 =–2(x+4) y– 7 =–2x – 8
y=–2x– 1
This can be converted to standard form by adding the quantity (2x+ 1) to both sides, simplifying, and rearranging:
y+2x+1=–2x – 1 +2x+1 2x+y+1=0
Quiz
This is an “open book” quiz. You may refer to the text in this chapter. You may draw diagrams if that will help you visualize things. A good score is 8 correct. Answers are in the back of the book.
1. Consider the following equation:
r=6 cos (q–π/4)
What does the graph of this equation look like in polar coordinates? (a) It is a straight line passing through the origin.
(b) It is a straight line that does not pass through the origin. (c) It is a circle centered at the origin.
(d) It is a circle that is not centered at the origin. 2. Consider the following equation:
Which of the following points lies on the graph of this equation in Cartesian coordinates? (a) (2,3) (b) (–2,3) (c) (2,–3) (d) (–2, –3)
3. In the graph of the equation 3x+y+5=0, what is the slope? (a) 3
(b) –3 (c) 5 (d) –5
4. In the graph of the equation y– 7 =0, what is the slope? (a) 7
(b) –7 (c) 0
(d) It is undefined.
5. Consider two points PandQon the Cartesian plane. Suppose that point Pis located at the origin, and the coordinates of point Qare (xq,yq). Now consider a third point Rwith coordinates (xr, yr), such that xr= 2xq and yr = 2yq. How does the distance dpqbetween points P and Q compare with the distance dprbetween points PandR?
(a) There is no way to tell without more information. (b) dpr=(dpq2+d
pr
2)1/2
(c) dpr=4dpq (d) dpr=2dpq
6. What is the distance din mathematician’s polar coordinates between the origin and the point q= π/ 2?
(a) d= π/ 2 (b) d= π
(c) d=(π/ 2)= π2/ 4
(d) This is a meaningless question, because in polar coordinates, q= π/2 does not represent a point.
7. What is the distance din mathematician’s polar coordinates between the origin and the point (q0,r0)=(–π/ 4,3)?
(a) d=3 (b) d= π/4 (c) d=5
(d) This is a meaningless question, because in polar coordinates, (q0,r0)
=(–π/ 4,3) does not represent a point.
8. Consider a town located halfway between the equator and the north geo- graphic pole. What is the longitude of this town?
(a) +45º (b) –45º (c) π/ 2
(d) There is no way to tell without more information.
9. Suppose a radar set displaying navigator’s polar coordinates indicates the presence of a hovering object directly northeast of us, and a range of 7 nautical miles. If we say that a nautical mile is the same as a “unit,” what are the coordinates (q0,r0) of this object in mathematician’s (not naviga- tors) polar coordinates, with q0expressed in radians?
(a) (π/ 4,7) (b) (π/ 2,7) (c) (3π/ 4,7) (d) (π,7)
10. Imagine a distant star that appears to pass, as the earth rotates, directly through the zenith point in the sky over a town located halfway between the equator and the north geographic pole. At a certain time (which varies over the course of a year) each day, the star appears at the zenith as observed from that town. What is the declination of this distant star? (a) +90º
(b) –90º (c) +45º