Since the 2 solids are similar and made from the same material, then their masses should be proportional to their surface areas:
[28 cm^2 / 40.32 cm^2] x 6,912 =4,800 grams - the mass of A
We are given that the solids are made from the same material, which means they have the same density.
Since mass = density * volume
massA / massB = (densityA * volumeA) / (densityB * volumeB)
densityA = densityB so we can cancel them out
massA / massB = volumeA / volumeB
So here we have shown that the ratio of the masses is equal to the ratio of their volumes.
Now we need to be careful and remember that the ratio of their volumes is not equal to the ratio of their surface areas.
The ratio of their volumes is equal to the ratio of their surface areas raised to the power of 3/2
volumeA / volumeB = ( SAA / SAB )3/2
Now we can replace volumeA / volumeB with massA / massB
massA / massB = ( SAA / SAB )3/2
Now let's plug in what we know for values of massB , SAA , and SAB
massA / 6912 = ( 28 / 40.32 )3/2
Multiply both sides by 6912 and simplify.
massA = 6912 * ( 28 / 40.32 )3/2
massA = 4000 and that is in grams
With any two similar objects A and B
If a pair of corresponding sides are in the ratio \(l_A:l_B\)
Pairs of corresponding Surface areas will be in the ratio \((l_A)^2:(l_B)^2\)
And for 3 dimensional objects the volumes will be in the ratio \((l_A)^3:(l_B)^3\)
Let a side of a hexagon be equal to the height (or length) of a solid.
Solid A A = 28 cm2
I'm gonna use this formula to calculate a2 ( and b2)
2(3√3*a2/2) + 6a2 = 28 a2 = 2.5008485 a = 1.581407127
V(A) = 3√3/2 * a = 10.27502675 cm3
Solid B A = 40.32 cm2
2(3√3*b2/2) + 6b2 = 40.32 b2 = 3.601221843 b = 1.897688553
V(B) = 3√3/2 * b = 17.75524625 cm3
V(B) : 6912 = V(A) : x
x = 4000 grams