Volume and surface area

[leonswati]

A certain object has a volume of 216 cubic centimeters and a surface area of 240 square centimeters. Another object is a smaller-scale replica of the original object. If the surface area of the smaller object is 60 square centimeters, what is its volume?

[mathbyvemuri]

As the formulae for S.A and Volume of the two objects are identical, the ratios of SA to Volume of the two objects are same.
V of smaller one/SA of smaller one = V of bigger one/SA of bigger one
=> V of smaller one/60 = 216/240
=> V of smaller one = 216*60/240 = 54

There is a flaw in this solution and the correct one is presented in my later post in this thread

[mathbyvemuri]

deleted

[leonswati]

The answer is 27

[mathbyvemuri]

yes there is a correction here:
to take care of the dimensions
V^(1/3)/SA^(1/2) ratios for the two objects must be equal
=> V^(1/3)/60^(1/2) = 216^(1/3)/240^(1/2)
=> V^(1/3) = 6/2 = 3
=> V = 3^3 = 27

The logic is here:
Unit of V is cm^3
Unit of SA is cm^2
In order to get the dimensionless ratio, we take ratio of cube root of V to the square root of SA.