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Area and Volume of Similar Solids

Solve problems using ratios between similar solids.

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Is There Life Out There?

One of the great mysteries of the world is whether or not life exists on other planets. Is it possible, or isn’t it? Astrophysicists have recently discovered three extrasolar planets where this possibility becomes much more likely. They have determined these planets to be habitable based on their similarities to Earth.

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These planets have been deemed habitable largely because of their temperatures, which fall in the ideal range for liquid water to exist on their surface. One of these three planets is Kepler-69c in the constellation Cygnus. Temperature-wise, Kepler-69c is similar enough to Earth that it could sustain life, but is it also mathematically similar to our planet?

As it turns out, Kepler-69c and Earth are also geometrically similar. They are both spheres, with Kepler-69c’s radius being 1.7 times larger than Earth’s.

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The ratio of the Earth’s radius to Kepler-69c's radius is 10:17.

  1. What is the ratio of Earth’s volume to Kepler-69c's volume?
  2. If Earth’s volume is \begin{align*}1.0975095 \times 10^{21}\end{align*} cubic meters, what is the volume of Kepler-69c?

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