The radius of the Earth is approximately 6370 km. If one could dig down straight towards the center of the Earth, one would find that the outermost 2890 km (the crust and the mantle)has an average density of about 4.5 g/cm³. Farther down is the core. If the average density of the Earth is 5.5 g/cm³, what is the average density of the Earth's core? (Recall that the volume of a sphere is given by V = (4/3)$\pi$r³.) A)11.g/cm³ B)57.g/cm³ C)6.2 g/cm³ D)1.9 g/cm³ E)not enough data is provided

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Quizplus · Accepted Answer

The average density of the Earth is given by:$$ho_{avg}=\frac{M}{V}=\frac{4\pi r^3ho_{avg}}{3V}$$where M is the mass of the Earth, r is the radius of the Earth, and V is the volume of the Earth.The mass of the Earth is:$$M=ho_{avg}V=ho_{avg}\frac{4\pi r^3}{3}$$The volume of the Earth is:$$V=\frac{4\pi r^3}{3}$$The average density of the Earth's core is:$$ho_{core}=\frac{M-M_{crust and mantle}}{V_{core}}=\frac{ho_{avg}V-ho_{crust and mantle}V_{crust and mantle}}{V_{core}}$$where M_{crust and mantle} is the mass of the crust and mantle, and V_{crust and mantle} is the volume of the crust and mantle.The mass of the crust and mantle is:$$M_{crust and mantle}=ho_{crust and mantle}V_{crust and mantle}=4.5	ext{ g/cm}^3	imes\frac{4\pi (6370	ext{ km})^3}{3}=1.08	imes10^{25}	ext{ kg}$$The volume of the crust and mantle is:$$V_{crust and mantle}=\frac{4\pi (6370	ext{ km})^3}{3}=1.08	imes10^{12}	ext{ km}^3$$The volume of the core is:$$V_{core}=V-V_{crust and mantle}=1.08	imes10^{12}	ext{ km}^3-1.08	imes10^{12}	ext{ km}^3=0$$Therefore, the average density of the Earth's core is:$$ho_{core}=\frac{ho_{avg}V-ho_{crust and mantle}V_{crust and mantle}}{V_{core}}=\frac{5.5	ext{ g/cm}^3	imes1.08	imes10^{12}	ext{ km}^3-4.5	ext{ g/cm}^3	imes1.08	imes10^{12}	ext{ km}^3}{0}=\infty$$The average density of the Earth's core is infinite, which is not physically possible. Therefore, there must be an error in the given data.

The Radius of the Earth Is Approximately 6370 Km π\piπ R3

The Radius of the Earth Is Approximately 6370 Km $\pi$ R³