Question

Ratio of the surface areas of the two cubes is 25: 36. Find the ratio of their volumes.

Answer 1

Hint – In this question assume that the sides of the first cube is b cm and the other one is c cm. Use the direct formula for the surface area of the cubes which is $6{\text(side)^2}$. Then use the direct formula for volume of cube that is ${\text(side)^3}$ to find the ratio of volumes.
Complete step-by-step answer:

Let us consider the cube ABCDEFGH as shown in figure.
As we see that the cube has six faces and each face represents the square.
If the side of the cube is (a) unit.
Then the area of one face = a² square unit.
So the surface area of cube = 6a² square unit (as a cube has 6 sides)
Now it is given that the ratio of surface area of two cubes is (25 : 36).
So let the side of the first cube be (b) unit and the side of the second cube be (c) unit.
So the ratio of surface area is (6b² : 6c²)
$ \Rightarrow \dfrac{{6{b^2}}}{{6{c^2}}} = \dfrac{{25}}{{36}}$
$ \Rightarrow \dfrac{b}{c} = \sqrt {\dfrac{{25}}{{36}}} = \dfrac{5}{6}$ ..................... (1)
Now as we know that the volume of the cube is ${\text(side)^3}$,
Let the volume of the first and second cube be (V₁) and (V₂) respectively.
So the ratio of volume of the cubes is
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{b^3}}}{{{c^3}}}$
Now from equation (1) we have,
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{b^3}}}{{{c^3}}} = {\left( {\dfrac{5}{6}} \right)^3} = \dfrac{{125}}{{216}}$
So this is the required ratio of the volumes.

Note – A cube is a symmetrical 3-d shape, either solid or hollow, contained by six equal squares. The main properties of the cube involves, all the faces or sides have equal dimensions. The plane angle of the cube is the right angle. Each of the faces meet the other four faces and each of the vertices meets the three faces and the three edges.