What is the mass of the continent? Answer in units of kg

Question

The current theory of the structure of the Earth, called plate tectonics, tells us that the continents are in constant motion. Assume that the North American continent can be represented by a slab of rock 5400 km on a side and 26 km deep and that the rock has an average mass density of 2890 kg/m3 . The continent is moving at the rate of about 2.4 cm/year. What is the mass of the continent? Answer in units of kg.

Answer 1

First it helps to understand that density is not the same thing as mass; it is the amount of mass that is present in a certain volume of space (mathematically: density = mass / volume). So if I tell you I have a piece of iron in my left hand, and a piece of foam sponge in my right hand, you might imagine that the iron is more massive than the foam, but what you actually know from experience is that the iron is more dense than the foam. If the iron piece were extremely tiny (like the size of a shirt button), while the foam piece was so big I could barely hold it in one hand, the foam could have a larger mass.

So using density = mass / volume, we would like to solve for the mass of the continent. We should rearrange the equation to solve for mass by multiplying volume to each side:

mass = density x volume.

The density of the continent is given as 2890 kg/m³ (a box 1 m on each side, like a typical big moving box, would contain 2,890 kg of a chunk of the continent, which is the mass of a typical family car).

The volume of the continent is hard to determine exactly since continents are very uniquely shaped, so the problem is having us make the assumption that is is a big square (5,400 km on each side) that goes 26 km deep. Since the density was given to us in kg/m³, we should calculate the volume in m³ so our units cancel when we plug everything into the equation.

The volume of a rectangular box is V = length x width x depth, so the volume of the continent is 5,400,000 m x 5,400,000 m x 26,000 m. This is equal to 7.581 x 10¹⁷ m³ (a HUGE volume, but it is, after all, a continent).

Plugging everything into our original equation:

mass = density x volume

mass = (2890 kg/m³) x (7.581 x 10¹⁷ m³)

mass = 2.19 x 10²¹ kg

This is a big number. But for context, the mass of the Earth is 5.97 x 1024 kg which is about 3,000 times bigger than the mass of the continent. This makes sense because the continents are all part of Earth's crust, which is a very thin layer on top of its core and mantle. So the answer makes sense.