The mean radius of the Earth is 6.37 ×10^6 m , and that of the Moon is 1.74 ×10^8 cm . From thes

step 1 Area ratio of art to moon is square of the radius ratio of art and mole, which is 6.37 times 10

The mean radius of the Earth is 6.37 ×10^6 m , and that of...

The mean radius of the Earth is $6.37 \times 10^{6} \mathrm{m}$ , and that of the Moon is $1.74 \times 10^{8} \mathrm{cm} .$ From these data calculate (a) the ratio of the Earth's surface area to that of the Moon and $(\mathrm{b})$ the ratio of the Earth's volume to that of the Moon. Recall that the surface area of a sphere is 4$\pi r^{2}$ and the volume of a sphere is $\frac{4}{3} \pi r^{3} .$

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Key Concepts

Unit Conversion

Ensuring unit consistency is critical in any calculation. When given values with different units, such as meters and centimeters, they must be converted into the same unit system to avoid errors in computed ratios or derived quantities.

Surface Area of a Sphere

This concept involves the formula 4?r², which is fundamental for determining the area covering a spherical object. Understanding how the surface area scales with the square of the radius is key when comparing two spheres.

Volume of a Sphere

The formula 4/3?r³ is used to calculate the volume enclosed by a sphere. This concept highlights the cubic relationship between the radius and the volume, which becomes particularly important when assessing scaling differences between objects.

Scaling Laws and Ratios

Scaling laws explain how geometric properties such as surface area and volume change with the size of an object. Specifically, while surface area scales with the square of the radius, volume scales with the cube, thus, differences in radius have a more pronounced effect on volume ratios than on area ratios. This principle is crucial when comparing the relative sizes of spherical bodies.

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