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Verified answer

Answer:

[tex]\boxed{\bf\: Surface\:area_{(Moon)} : Surface\:area_{(Earth)} = 1 :16 \: } \\ [/tex]

Step-by-step explanation:

Given that, diameter of the moon is approximately one fourth of the diameter of the earth.

Let assume that

[tex] \sf \: Diameter\:of\:earth = x \: units \\ [/tex]

So,

[tex] \sf \: Diameter\:of\:moon = \dfrac{x}{4} \: units \\ [/tex]

Thus,

[tex] \sf \: Radius\:of\:earth = \dfrac{x}{2} \: units \\ [/tex]

[tex] \sf \: Radius\:of\:moon = \dfrac{x}{8} \: units \\ [/tex]

Now, Consider

[tex]\sf\: Surface\:area_{(Earth)} : Surface\:area_{(Moon)} \\ [/tex]

[tex]\sf\: = \: 4\pi \: (Radius_{(Earth)})^{2} : 4\pi \: (Radius_{(Moon)})^{2} \\ [/tex]

[tex]\sf\: = \: (Radius_{(Earth)})^{2} : (Radius_{(Moon)})^{2} \\ [/tex]

[tex]\sf\: = \: \bigg( \dfrac{x}{2} \bigg) ^{2} : \bigg( \dfrac{x}{8} \bigg) ^{2}\\ [/tex]

[tex]\sf\: = \: \dfrac{ {x}^{2} }{4} : \dfrac{ {x}^{2} }{64} \\ [/tex]

[tex]\sf\: = \: 1 : \dfrac{ 1 }{16} \\ [/tex]

[tex]\sf\: = \: 16 : 1 \\ [/tex]

Hence,

[tex]\implies\boxed{\bf\: Surface\:area_{(Earth)} : Surface\:area_{(Moon)} = 16 :1 \: } \\ [/tex]

Or

[tex]\implies\boxed{\bf\: Surface\:area_{(Moon)} : Surface\:area_{(Earth)} = 1 :16 \: } \\ [/tex]