About cube: One cube has a face equivalent to the area of another cube,find the ratio of their volumes.?

Question

* Each of the six faces of a cube is a square of edge a. Since the area of one face is=a squared,the total area of the cube is 6a squared.

*FORMULA

for the total area:
At=6(area of one face)
=6a squared
for volume:
V=(edge)cube
=a cube

Can u give me the complete solution for this answer?
Actually the answer is square root of 6 over 36.

Answer 1

Take what you know, surface area of cube a = area of one side of cube b, into equation:
6a^2 = b^2
Then solve for one by squarerooting both sides:
6^(1/2)a = b (6^(1/2) is the square root of 6)
Now, cube both sides to show volume:
b^3 = 6^(3/2)a^3
The ratio will be the coefficients of the variables:
1 : 6^(3/2)
1 : 6^(1/2) * 6^(2/2) Subtract 2/2 (or 1) from exponent
1 : 6*6^(1/2) 6^(2/2) = 6^1 = 6
6^(1/2) : 6*6(1/2)*6^(1/2) The authors don't want square root in denominator, so mult. both sides by square root of 6
6^(1/2) : 6^2 when multiplying same base you can add exponents 1 + 1/2 + 1/2 = 2
6^(1/2) : 36 This is square root of 6 over 36

Answer 2

Cube1 has volume a^3 and surface area 6a^2
Cube 2 has volume A^3 and surface area 6A^2
What is relationship between a (edge of smaller cube) and A (edge of larger cube)?
A^2 (face) = 6a^2 (face)
sqrt(A^2) = sqrt(6a^2)
A = a*sqrt6
ratio of the areas = A^3/a^3 = (a*sqrt6)^3 / (a^3) =
a^3* (sqrt6)^3 / (a^3) =
(sqrt6)^3 = sqrt216 = 6 sqrt6
This is the ratio of the larger to the smaller

The answer you show is the ratio of the smaller to the larger.
1/6sqrt6 = sqrt6 / 6sqrt*sqrt6 = sqrt6 / 36