Question Video: Solving Problems Involving the Density of a Sphere Mathematics

Earth has a mass of 1.317 × 10²⁵ lb and a radius of 3959 miles. By modeling Earth as a perfect sphere, work out its density to the nearest pound per cubic foot, knowing that density = mass/volume.

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Video Transcript

Earth has a mass of 1.317 times 10 to the 25th power pounds and a radius of 3959 miles. By modelling Earth as a perfect sphere, work out its density to the nearest pound per cubic foot, knowing that density equals mass over volume.

So in this question, we need to work out the density of Earth, using the formula density equals mass over volume. We’re given the mass of Earth, but not the volume. We can, however, work it out given that we’re told to model Earth as a sphere and we’re given the radius. So here, we have Earth modelled as a sphere, and we can use the formula that the volume of a sphere equals four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius. So substituting in the value for the radius of 3959 miles, we would have the volume is equal to four-thirds 𝜋 times 3959 cubed. Using our calculator, we can evaluate this as 2.599232 and so on times 10 to the 11th power cubic miles.

If we look at the question, however, we can see that we’re asked for density in terms of the nearest pound per cubic foot. So this means that we don’t want our volume in terms of cubic miles, but instead in terms of cubic feet. We can use the conversion that one cubic mile is equal to 1.472 times 10 to the 11th power cubic feet. So we can take the volume of Earth in cubic miles and multiply it by the value 1.472 times 10 to the 11th power to get a value in cubic feet, which is 3.826070116 and so on times 10 to the power of 22 cubic feet. We won’t round this value yet, since it will give us the most accurate final answer. We can now work out the density of Earth using the given value of mass and the volume we’ve just calculated.

Therefore, the density of Earth is equal to 1.317 times 10 to the 25th power over 3.82607116 and so on times 10 to the 22nd power, which is 344.2174242 and so on. And since we’ve divided a unit in pounds by a unit in cubic feet, then the units of our density will be pounds per cubic feet. And one final part remains, and that is to round our answer to the nearest pound per cubic feet. So we check our first decimal place digit to see if it’s five or more. And as it’s not, this means our answer stays as 344 pounds per cubic feet, which is our final answer for the density of Earth.

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