Question

The diameter of earth is four times (approximately) the diameter of the moon, then the ratio of their surface area is?
(A). $4:1$
(B). $8:1$
(C). $16:1$
(D). $64:1$

Answer 1

Hint- In this question use the formula for surface area of the sphere because earth is in the form of a sphere, which is given as $4\pi {r^2}$. The radius can easily be obtained using diameters so use the given relation between the diameter of earth and moon to get the answer.


Complete step-by-step solution -
We have given every value given but the first basic concept is the value of the diameter of earth is given as 4 times that of the moon.
And remember we have to give the ratio of earth to that of the moon which is clear by options but may or may not be given in examinations.
Coming back to the question another thing to know is the main thing the surface area of sphere (remember sphere is the closest shapes these celestial objects can be related) which is $4\pi {r^2}$ by this concept your question will be done also remember to take the values as per given in question and to differentiate between them.
Let earth’s diameter be ‘D’ and moon’s diameter be ‘d’
Similarly, let earth’s radius be ‘R’ and moon’s radius be ‘r’
Given earth’s diameter = $4 \times $ moon’s diameter
$
   \Rightarrow D = 4d \\
   \Rightarrow \dfrac{D}{2} = 2d \\
   \Rightarrow R = 4r \\
 $
Ratio of surface areas,
$
   \Rightarrow 4\pi {R^2} = 4\pi {r^2} \\
   \Rightarrow {\left( {4r} \right)^2} = {r^2} \\
   \Rightarrow 16{r^2} = {r^2} \\
 $ (using surface area of sphere $4\pi {r^2}$)
By cancelling ${r^2}$ we get $16:1$.
Hence, (C) is the correct option.

Note- In this question the major mistake that can be done is the interpretation of the shape of earth and moon. It could easily be mistaken as elliptical or oval. We need to understand that earth being a 3-d circle comes in the category sphere while solving such mathematics questions.