In class I mentioned that the ratio of the surface area to the volume of a plane

Question

In class I mentioned that the ratio of the surface area to the volume of a planet affects how fast it can cool. For a sphere, the surface area is A = 4 pi r^2 and the volume is V = (4/3) pi r^3, where r is the radius of the sphere. The radius of Mars is roughly two times larger than the radius of the Moon. (a) Calculate the ratio of the surface areas of the two objects: A_Mars/A_Moon (b) Calculate the ratio of the volumes of the two objects: V_Mars/V_Moon (c) As we discussed in class, and as you saw on the midterm, you can plug in the expressions for A and V to find out that A/V = 3/r Use this to calculate (A/V)_Mars/(A/V)_Moon (d) Based on your answer for part (c), did the Moon or Mars cool off faster?

Explanation / Answer

the radius of Mars is R = 3390*10^6 m

Radius of moon is r = 1737*10^6 m

given these are of spherical shape so the

surface area and volume of them is 4piR^2 , (4/3)piR^3

          
   moon is    4pir^2, (4/3)(pir^3)

a)
now the ratio of surface area of the two objects is A/a = 4piR^2/4pir^2

                           = R^2/r^2

                           = (3390*10^6)^2 /(1737*10^6)^2

                           = 3.80890
b)

the ratio of volume is V/ v = (4/3)piR^3/(4/3)pir^3

           = R^3/r^3

           = (3390*10^6)^3 /(1737*10^6)^3
          
           = 7.43361

c) ratio of A/V = 3/R

       (A/V)Mars /(A/V)moon = (3/R)/(3/r) = r/R = (1737*10^6)/(3390*10^6) = 0.512389380

d) Mars will cool faster than moon

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