Question

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

please explain step by step

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Answer 1

Given :

• Diameter of the moon = ¼ of the diameter of the Earth.

To find :

Ratio of the surface area of moon and the surface area of earth.

Solution :

Let the diameter of the Earth be x m.

Thus according to the Question , diameter of the moon will be ¼ of x.

We know the formula for surface area of a sphere i.e,

$\boxed{\bf{TSA = 4\pi r^{2}}}$

Where ,

• TSA = Total surface area of the sphere

• r = Radius of the sphere

Surface area of the Earth in terms of x (Radius) :

Using the formula for total surface area of a sphere and substituting the variable x in it , we get :

$:\implies \bf{TSA = 4\pi r^{2}}$

$:\implies \bf{TSA = 4\pi x^{2}}$

$\boxed{\therefore \bf{TSA = 4\pi x^{2}}}$

Hence the surface of the Earth (in terms of x) is 4π x².

Surface area of the moon :

Using the formula for total surface area of a sphere and substituting the variable x in it , we get :

$:\implies \bf{TSA = 4\pi r^{2}}$

$:\implies \bf{TSA = 4\pi \times \dfrac{1}{4}x^{2}}$

$:\implies \bf{TSA = \not{4}\pi \times \dfrac{1}{\not{4}}x^{2}}$

$:\implies \bf{TSA = \pi x^{2}}$

$\boxed{\therefore \bf{TSA = \pi x^{2}}}$

Hence the surface of the moon (in terms of x) is π x².

Now , by dividing the surface area of moon by surface area of earth , we get :

$:\implies \bf{\dfrac{TSA_{m}}{TSA_{e}} = \dfrac{\pi x^{2}}{4\pi x^{2}}}$

$:\implies \bf{\dfrac{TSA_{m}}{TSA_{e}} = \dfrac{\not{\pi} x^{2}}{4\not{\pi} x^{2}}}$

$:\implies \bf{\dfrac{TSA_{m}}{TSA_{e}} = \dfrac{x^{2}}{4x^{2}}}$

$:\implies \bf{\dfrac{TSA_{m}}{TSA_{e}} = \dfrac{\not{x^{2}}}{4\not{x^{2}}}}$

$:\implies \bf{\dfrac{TSA_{m}}{TSA_{e}} = \dfrac{1}{4}}$

$\boxed{\therefore \bf{TSA_{m} : TSA_{e} = 1 : 4}}$

Hence the ratio of the surface area of moon and the surface area of Earth is 1 : 4.