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An important question to consider when thinking about global warming is, "If the ice sheets near the poles melt, how much will the sea level rise?" This seems like a difficult question, given the odd shapes of both the ice sheets and the oceans. But there are some accurate approximations that allow the answer to be estimated fairly accurately with reasonably simple calculations.

The critical idea is that both the thickness of the ice sheets and the amounts of the sea-level rise are extremely small compared to the radius of the earth. The radius of the earth is about 2/π x 10⁷ m -- more than 6000 kilometers. The ice sheet thicknesses we will be concerned with are single digit miles and the sea level rises will be in dozens of feet. As a result, in thinking about them, we can essentially ignore the curvature of the earth. We can imagine peeling the map of the earth off a globe and flattening it out (by making cuts, not by stretching it, so that we preserve the area). Then, both the ice and the sea level rise can be treated as right (not tilted) cylinders (though with funny shaped bases and tops). Since we know that the volume of a right cylinder is the area of the base times the height, we can easily estimate all the volumes we need. A schematic picture of this approximation (with the height of the ice mass greatly exaggerated -- you couldn't see it if I didn't) is shown below. The error in these approximations is on the order of the height of the cylinder considered divided by the radius of the earth; a very small number.

a. Assume that (after flattening the surface of the earth in our imaginations) the ice sheet to be considered covers an area, A, and has a thickness, d. The oceans currently cover about 75% of the earth's surface, and note that the surface area of a sphere is 4πr2 where r is the radius of the sphere. Generate an equation that will allow you to calculate h, the height the ocean levels will rise due to the melting of a an ice sheet in terms of A, d, and r b. As shown in the combined satellite photo on the right (from Google Earth), Greenland is covered by a sheet of ice. This ice has been measured to have a mean thickness of about 2 km. Recent observations indicate that this ice sheet is beginning to retreat -- that it is melting at an accelerating rate. Using the depth of the ice sheet and the scale given on the picture, estimate how much sea level rise would be produced by the melting of the entire ice sheet lying on Greenland. (Note: You can easily look this information up on the web. But the goal of this problem is in part to develop your estimation skills for the purpose of building your ability to decide for yourself whether any given piece of information you find on the web is reasonable or bogus.) c. Find a website that allows you to see what the coast of Florida would look like if the sea level rose the amount you calculated. Include a printout of this picture as part of your solution and give a brief reason why you trust this site.

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Page last modified February 3, 2010: G27