3 Calculating the Mass of the Earth The interior structure of the Earth is very difcult to probe: we live on the very surface of the crust, a rocky layer over 20 miles thick, and to date the deepest humans have ever drilled is the Kola Superdeep Borehole, dug to a. depth of 7.69 miles by the USSR, (and stopped when the heat. and pressure was melting through all drillbits). Thus, it is a. true wonder of modern science that we are able to calculate the density of the earth at various depths without directly retrieving samples: instead, we use the vibrations of earthquakes (and a lot of sophisticated mathematics) to deduce the density deep below the crust. To a. very good approximation, the earth is spherically symmetric], and so we may record the density of the earth as a. function p[r] of the radius r from the center. With the density function ,0 in hand, the mass of the earth E is given by the triple integral EU = III pdV E \"Fhile the true density function of the earth is rather complicated, its general behavior is well-approximated by the following function: its more dense in the center at the metal core, with density dropping off to an approximate constant level in the mantle and surface (where the earth is mostly rock, not metal): b 1 + 3'2 \"There r is measured in thousands of kilometers and a. is the density of rock (a = 3. llllfl'kgf n13), and b is the difference in density between rock and metal, or b = 5. kgf'mg. For the coming calculations, its useful to compute these to the units of kitogrums per cubic I 190ka to match the units on r: That is a = 3- 113:12 and b = 6-10\"). p(r) = (1+ Approximating the earth as a. sphere of radius R = 6. 000 kilometers: set up a triple integral in spherical coordiantes that nds the mass of the earth. Solve this integral (don't plug in the numbers until the end: as the formula will be much cleaner if you keep R. a, b as variables), and then plug in the relevant constants. \""hat is the mass of the earth in kilograms? (Plug the numbers in using a calculator or computer: the result will be a. very large number: best reported in scientic notation!)