Page 484  No. 2:

    From Kepler's Third law, (harmonic law), the square of the sidereal period is proportional to the cube of its semimajor axis (one-half of the major axis)

            (period) squared varies as (semimajor axis) cubed  The constant of proportionality is 1 (year) squared divided by (AU) cubed.

No. 2.    semi major axis = 4 AU.          Its cube is 64 AU cubed  (AU) cubed cancels
                So period squared is 64 years squared
                Hence period is 8 years.
 

Page 484 No. 4:

    From Newton's correction to Kepler's third law:

    The square of the period equals (4 times square of PI times cube of distance to Sun) divided by (the sum of the Mass of the planet and the sun) multiplied by the gravitational constant (which equals 6.67 X 10^-11)

No. 4.    All the terms are constants except the period and the mass of the Sun.  We can ignore the mass of the Earth.        So  if the mass is of the Sun is four times the period will be the reciprocal of the square root of 4 which is 0.5  Hence the period of revolution will be Half a year.
 

Page 509/510

No. 2.    The angle is 105 - 75 = 40 degrees.

No. 4.    At 60 nautical miles per degree, the distance is 60 times 40 which equals 2400 nautical miles.

No. 6.    East of your starting point.

No. 8.    36 degrees South, 40 degrees West

No. 10.    110 degrees - 70 degrees equals 40 degrees.  At 15 degrees per hour and 4 minutes per degree this equals 2 hours and 40 minutes.  1 A.M. - 2 hours 40 minutes is 10:20 P.M. the previous day October 15.

No. 12.    For Standard Time, use the nearest multiple of 15.  118 becomes 120 W and 140 becomes 135 E the difference is 255 which is 17 hours.  Adding 17 to 10 AM gives 27 which is 3 A.M. on the next day February 23.

No. 14.    5 P.M.

No. 16.    As in No. 12, the difference from 75 W and 15 E is 90 degrees or 6 hours.  So the time is 7 P.M.

No. 18.    From Table 17.1 page 500, at 40 N, there are 9 hours of daylight on December 21 and 15 hours on June 21, that is 6 hours difference or 1 hour per month.  Interpolating on April 21, there will be 13 hours of sunlight.  Maximum altitude at Noon.  Half before, so 6.5 hours before noon. So Sunrise is at 5:30 A.M.

No. 20.    (a)    The zenith angle is 34 degrees.  Altitude is 90 - 34 = 56 degrees.

                (b)     The zenith angle is 34 + 23.5 = 57.5 degrees.  Altitude is 90 - 57.5 = 32.5 degrees.

No. 22.    Altitude is 31.5 degrees.  So Zenith angle is 58.5 degrees.
                Latitude + 23.5 = zenith angle.  so Latitude is 58.5 - 23.5 = 35 degrees.

No. 24.    The Sun is at minimum altitude on December 22.  Zenith angle is 39 + 23.5 = 62.5.
                Altitude = 90 - 62.5 = 27.5 degrees.

Page 534/535

No. 2.    120/6  =  20 pounds.

No. 4.    12 multiplied by 29.5 = 354 days.

No. 6.    (a)    36 - 23.5 = 12.5.  12.5 - 5 = 7.5.  90 - 7.5 = 82.5 degrees

              (b)    This occurs on December 22.

No. 8.    (a)    33.5 - 23.5 = 10.  10 - 5 = 5.  90 - 5 = 85 degrees.

             (b)    This occurs on September 21.

No. 10.    (a)    From Page 524, Table 18.1, The last quarter moon is overhead at 6.00 A.M.  If the observer sees it overhead at 12 Noon, the observer must be 6 hours ahead, which is 90 degrees to the East.  So the place must be 120 - 90 which is 30 degrees West.

                (b)    The last quarter moon rising is at Midnight.  So the observer' time must be Noon, and the longitude exactly 180 degree from 120 W which is   60 degrees East.

                (c)    The last quarter moon setting is at Noon.  So the longitude is 120 degrees West and the Time is Midnight.

No. 12.    (a)    West.

                (b)    last-quarter phase; east (right) side.

No. 14.    39 - 23.5 = 15.5 - 5 = 10.5.  90 - 10.5 = 79.5   on December 21.

No. 18.    (a)    180 - 118 = 62 degrees East.

                (b)    118 - 90 = 28 degrees West and 62 + 90 = 152 degrees East

Pages 571/572

No. 2.    (4 X 10^31 X 92% X 15%)/(3 X 10^13 X 3600 X 24 X 365.2422) = 5.83 billion years.

No. 4.    The reciprocal of 0.2 which is 5 parsecs.

No. 6.    300,000,000 X 3600 X 24 X 365.2422 = 9.467 X 10^15 meters.

No. 8.    4.3/3.26 = 1.319 parsecs.

No. 10.    (a)    1000    (b)    5000

No. 12.    (2 X 6.67 X 10^-11 X 2 X 10^32 X 10^-16)/9 = 296.444 meters or 296.44 kilometers.

No. 14     5 billion divided by 200,000,000 = 25 times.

No. 16.    (3.086 X 10^19)/(75 X 3.16 X 10^7) = 13 billion years

Page 718.

No. 4.    10,000 meters equals 10 kilometers.  At 6.5 degrees per kilometer, the decrease is 65 degrees C.  20 - 65 = -45 degrees Celsius.

No. 6.    Use Tables A.1 and A.2 on pages A14 and A15.   The difference in temperatures is 95 - 90 = 5 degrees F.

            (a)    Relative Humidity (R.H.) is 82%

            (b)    Dew Point is 89 degrees F.

            (c)    Maximum moisture Content (M.C.) is 17.1 grams per cubic feet.

            (d)    Actual moisture content of the air (A.C.) is 17.1 times 82% = 14.02 grams per cubic foot.

No. 8.      Use Tables A.1 and A.2 on pages A14 and A15.   The difference in temperatures is 95 - 90 = 6 degrees F.

            (a)    R.H. is 45% and M.C. is 2.4.  So A.C. = 45% * 2.4 = 1.08 grams per cubic foot.

            (b)    The dew point is 17 degrees.  So it would not freeze.

No. 10.    From Table A.1, the difference in temperatures for a R.H. of 79% with an air temperature of 80 F is 5 degrees F.  From Table A.2, the dew point is 73 Degrees.  So the air temperature must be lowered by 80 - 73 = 7 degrees F.

Page 748.

No. 6.    11/5 = 2.2 miles.  or 11/3 = 3.67 kilometers