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The diameter of the Moon is 3480 km. (a) What is the surface area of the Moon? (b) How many...

Question:

The diameter of the Moon is 3480 km.

(a) What is the surface area of the Moon?

(b) How many times larger is the surface area of the Earth?

Solar system

The Solar System is formed by celestial bodies held in place by the gravitational attraction of the Sun. The principal components of the Solar System are the eight planets forming it, although there are several other celestial bodies (moons, asteroids, etc) also moving in the orbit of the Sun.

Ordering the planets attending to their size:

{eq}\bullet {/eq} Jupiter ({eq}69,911 \;\rm km {/eq}), over ten times the size of Earth.

{eq}\bullet {/eq} Saturn ({eq}58,232 \;\rm km) {/eq}, 9.5 times the size of Earth

{eq}\bullet {/eq} Uranus ({eq}25,362 \;\rm km {/eq}), 4 times the size of Earth

{eq}\bullet {/eq} Neptune ({eq}24,622 \;\rm km {/eq}), 3.9 times the size of Earth

{eq}\bullet {/eq} Earth {eq}(6,371 \;\rm km) {/eq}.

{eq}\bullet {/eq} Venus {eq}(6,052 \;\rm km) {/eq}, 0.95 times the size of Earth.

{eq}\bullet {/eq} Mars {eq}(3,390 \;\rm km) {/eq}, 0.53 times the size of Earth.

{eq}\bullet {/eq} Mercury {eq}(2,440 \;\rm km) {/eq}, 0.38 times the size of Earth.

{eq}\bullet {/eq} Moon ({eq}1740\;\rm km {/eq}), 0.27 times the size of Earth.

Answer and Explanation:

The Moon is modeled as a sphere with radius,

{eq}R_M=\dfrac{D_M}{2}=\dfrac{3480\;\rm km}{2}=1740\;\rm km {/eq}.

a) The surface of the Moon is computed from the formula for the surface of a sphere,

{eq}S_{Moon}=4\pi R_M^2 {/eq},

evaluating the expression,

{eq}S_{Moon}=4\cdot 3.1416 \cdot 1740^2\;\rm km^2=\boxed{3.80\times 10^7 \;\rm km^2} {/eq}.

b) Let us compare the surface of the Moon with that of the Earth. The Earth is modeled as a sphere with radius,

{eq}R_E=6371\;\rm km {/eq}.

The Earth's surface is,

{eq}S_E=4\pi R_E^2=4\cdot 3.1416\cdot 6371^2\;\rm km^2=5.10\times 10^8 \;\rm km^2 {/eq}.

Dividing the surface of the Moon by the surface of the Earth,

{eq}\eta=\dfrac{S_M}{S_E}=\dfrac{3.80\times 10^7 \;\rm km^2}{5.10\times 10^8 \;\rm km^2}=\boxed{0.075} {/eq}.

The ratio is {eq}\eta=0.075 {/eq}, consequently, the surface of the Earth is {eq}\boxed{13.33} {/eq} times larger than the surface of the Moon.


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