Question

The diameter of the moon is approximately one-fourth of the diameter of the earth. Find the ratio of their surface areas.
(a) 1:16
(b) 16:1
(c) 1:8
(d) 8:1

Answer 1

Hint: We are given the diameter of the moon is one-fourth of the earth's diameter. We will first assume the diameter of the earth be x. So, we get the moon diameter is \[\dfrac{1}{4}\] of x. Now, after this, we will cover this diameter into the radius using \[r=\dfrac{d}{2}.\] At last, we will find the surface area of the moon and the earth using \[S=4\pi {{l}^{2}}\] and then find the ratio.

Complete step-by-step solution:
We are given that the moon’s diameter is one – fourth of the diameter of the earth. We will assume that the earth's diameter is x meter. Then, as by the above-given statement, the diameter of the moon will be one – fourth of x. So, the diameter of the moon will be,
\[{{D}_{m}}=\dfrac{1}{4}\text{of }x\]
\[\Rightarrow {{D}_{m}}=\dfrac{x}{4}m\]
So, we have the earth’s diameter, \[{{D}_{e}}=x\] and the Moon’s diameter is \[{{D}_{m}}=\dfrac{x}{4}.\]
Now, we are asked to find the surface area of the ratio of the moon with the earth. The earth and the moon both are in the shape of the sphere.
The surface area of the sphere is given by the formula, \[S=4\pi {{r}^{2}}\] where r is the radius.
Now, for the moon, we know that the radius is half of the diameter. So,
\[{{r}_{m}}=\dfrac{{{d}_{m}}}{2}\]
Hence, we have,
\[\Rightarrow {{r}_{m}}=\dfrac{\dfrac{x}{4}}{2}\]
\[\Rightarrow {{r}_{m}}=\dfrac{x}{8}\]
Now, the surface area of the moon is given as
\[{{S}_{m}}=4\pi {{r_m}^{2}}\]
\[\Rightarrow {{S}_{m}}=4\pi {{\left( \dfrac{x}{8} \right)}^{2}}\]
\[\Rightarrow {{S}_{m}}=\dfrac{4\pi {{x}^{2}}}{64}\]
Now, the earth radius is half of the diameter. So,
\[{{r}_{e}}=\dfrac{{{D}_{e}}}{2}\]
\[\Rightarrow {{r}_{e}}=\dfrac{x}{2}\]
Now, the surface area is given as,
\[{{S}_{e}}=4\pi r_{e}^{2}\]
Putting, \[{{r}_{e}}=\dfrac{x}{2},\] we get,
\[\Rightarrow {{S}_{e}}=4\pi {{\left( \dfrac{x}{2} \right)}^{2}}\]
\[\Rightarrow {{S}_{e}}=\dfrac{4\pi {{x}^{2}}}{4}\]
Now, we have both of their surface areas. So,
\[\text{Ratio of their surface area}=\dfrac{\text{Surface Area of Moon}}{\text{Surface Area of Earth}}\]
\[\Rightarrow \text{Ratio of their surface area}=\dfrac{{{S}_{m}}}{{{S}_{e}}}\]
As, \[{{S}_{e}}=\dfrac{4\pi {{x}^{2}}}{4}\] and \[{{S}_{m}}=\dfrac{4\pi {{x}^{2}}}{64},\] so we get,
\[\Rightarrow \text{Ratio of their surface area}=\dfrac{\dfrac{4\pi {{x}^{2}}}{64}}{\dfrac{4\pi {{x}^{2}}}{4}}\]
Now, cancelling the like terms, we get,
\[\Rightarrow \text{Ratio of their surface area}=\dfrac{\dfrac{1}{64}}{\dfrac{1}{4}}\]
\[\Rightarrow \text{Ratio of their surface area}=\dfrac{1}{64}\times \dfrac{4}{1}\]
\[\Rightarrow \text{Ratio of their surface area}=\dfrac{1}{16}\]
Hence, the required ratio is \[\dfrac{1}{16}=1:16.\]
Therefore, the correct option is (a).

Note: The surface area of the sphere is defined as \[4\pi {{r}^{2}}.\] Also, remember we have \[{{r}_{e}}=\dfrac{x}{2}\] then \[{{\left( {{r}_{e}} \right)}^{2}}=\dfrac{{{x}^{2}}}{{{2}^{2}}}=\dfrac{{{x}^{2}}}{4}.\] Sometimes we miss squaring the denominator and use \[{{r}^{2}}=\dfrac{{{x}^{2}}}{2}\] only which will lead us to the wrong answer.