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The diameter of the moon is approximately one fourth of the diameter of the earth. The ratio of their surface areas is ____.

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Detailed Solution

Concept-
We are aware that the sphere's surface area is 4 π

r^{2}

. The radius of a sphere is r.
Let earth's diameter equal x.
The moon's diameter is one-fourth that of the earth's.
So, moon diameter =

\frac{x}{4}

So, to calculate radius:
earth's radius Equals

\frac{x}{2}

Moon's radius is

\frac{1}{2} \times \frac{x}{4}

, or

\frac{x}{8}

.
Now, their surface area ratio equals moon surface area to earth surface area.
4π is being cancelled from both the numerator and the denominator. We obtain: =

\frac{(Moon Radius) * 2}{(Radius of earth) * 2}

Substituting the moon's and the earth's radius. We obtain =

\frac{{(\frac{x}{8})}^{2}}{{(\frac{x}{2})}^{2}}

If you apply the denominator term to the numerator, you get =

\frac{x^{2}}{8^{2}} \times \frac{2^{2}}{x^{2}}

\frac{4}{64}

.
By eliminating the

x^{2}

term, we arrive at the result: = 464 (Simple division) =

\frac{1}{16}

.
Therefore, the moon's surface area to that of the earth is 1:16.
The right response is thus 1:16.

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