Question

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

Answer 1

Hint: We solve this by taking the diameter of earth. dividing the diameter by two, we get the radius of earth. Similarly by using the given condition we can find the radius of the moon. Since earth and moon look like a sphere. By using the area of the sphere we can find the ratio of the moon and earth surface area.

Complete step-by-step answer:
We know that the surface area of the sphere is \[ = 4\pi {r^2}\] . \[r\] Is the radius of a sphere.
Let diameter of earth \[ = x\]
Diameter of the moon is one fourth the diameter of the earth.
So, diameter of moon \[ = \dfrac{x}{4}\]
Hence, to find radius:
Radius of earth \[ = \dfrac{x}{2}\]
Radius of moon \[ = \dfrac{1}{2} \times \dfrac{x}{4}\] \[ = \dfrac{x}{8}\] .
Now,
Ratio of their surface areas \[ = \dfrac{{{\text{surface area of moon}}}}{{{\text{surface area of earth}}}}\] .
 \[ = \dfrac{{4\pi {{({\text{Radius of moon)}}}^2}}}{{4\pi {{({\text{Radius of earth)}}}^2}}}\] .
Cancelling, \[4\pi \] on the numerator and on the denominator. We get:
 \[ = \dfrac{{{{({\text{Radius of moon)}}}^2}}}{{{{({\text{Radius of earth)}}}^2}}}\] .
Substituting, radius of moon and earth. We get,
 \[ = \dfrac{{{{\left( {\dfrac{x}{8}} \right)}^2}}}{{{{\left( {\dfrac{x}{2}} \right)}^2}}}\]
 \[ = \dfrac{{\dfrac{{{x^2}}}{{{8^2}}}}}{{\dfrac{{{x^2}}}{{{2^2}}}}}\]
Taking denominator term to the numerator,
 \[ = \dfrac{{{x^2}}}{{{8^2}}} \times \dfrac{{{2^2}}}{{{x^2}}}\]
 \[ = \dfrac{{{x^2}}}{{64}} \times \dfrac{4}{{{x^2}}}\]
Cancelling the \[{x^2}\] term, we get:
 \[ = \dfrac{4}{{64}}\] (Simple division)
 \[ = \dfrac{1}{{16}}\] .
Thus, the ratio of the surface area of the moon to earth is 1:16.
So, the correct answer is “1:16.

Note: If you want to apply the surface area of some objects, you need to know the shape of that object. In this case both the moon and earth are spherical in shape. Hence we apply the surface area of the sphere in both cases. While finding the radius of the moon be careful because we need to divide the diameter \[\dfrac{x}{2}\] by 2.