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If the Terrestrial Poles Were to Melt...
First, let us estimate the total volume of the south polar ice. We begin by noting that, on the Earth, 1 degree of latitude corresponds to a distance of 69.2 mi. Now, using a well marked map, Antarctica may be estimated as having an angular radius of 18 degrees of latitude, or a radius in miles of rA = 69.2 mi/deg x 18 deg = 1.25 x 103 mi. Its estimated surface area is then rA2 = 4.91 x 106 mi2. If we assume the average thickness of the Antarctic ice to be 1 mi, then the total ice volume is
VIce = 4.91 x 106 mi3.
If this ice melts, the resulting volume of liquid water is 4/5 of this value or
VWater = 3.93 x 106 mi3.
Now, let this volume of water be spread in a uniform, thin shell over a uniform Earth (for the moment, we will ignore the continents). The total surface area of the Earth is AE = 4rE2 = 1.97 x 108 mi2, where rE = 3.96 x 103 mi. Since the thickness of this shell, rE, may be assumed to be << the Earth's radius, rE , we may write
3.93 x 106 mi3 AErE1
or, rE = 1.99 x 10-2 mi = 105 ft.
If we now return the continents to their proper place, it is easy to see that the value of rE must become larger. Since only 2/3 of the earth's surface area is ocean, the thickness of the water shell must be increased by a factor of about 3/2:
rE = (3/2) x 105 ft = 158 ft. 2
Thus, we have estimated that if the polar ice were to melt, the Earth's oceans would rise by about 150 ft. Reference to a map showing land elevations allows us to determine that coastal areas and major river valleys (such as the Mississippi or the Amazon river valleys) would be flooded. The land distribution would appear somewhat differently than it does on the globe today. But the majority of land would still be above sea level.
Actually the volume of the water shell is VWater = (4/3)(r23 - r13), where r1 is the radius of the Earth prior to flooding (3.96 x 103 mi), and r2 is the radius of the Earth after flooding (r2 = r1 + rE). We may simplify the above expression for VWater to read VWater = (4/3)(rE3), where (rE3) = (r1 + rE)3 - r13= 3r2rE + 3r1(rE)2 + (rE)3 3r2rE, since rE << r1. Thus VWater 4r2rE = AErE. This last expression is the one given in the text.
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