Question

The ratio of surface areas of two cubes is 4 : 9. The ratio of their volumes is ______
(a) 2:3
(b) 64 :27
(c) 27 : 64
(d) 8 : 27

Answer 1

Hey there!!

The formula for the surface area of a cube is 6a²

a = side

Let the side of 1st cube be a

The side of the second cube be b

6a² / 6b² = 4 / 9

a² / b² = 4 / 9

( a / b )² = ( 2 / 3 )²

a / b = 2 / 3

Side of 1st cube = 2

Side of second cube = 3

Volume of cube = ( side )³

a / b = 2 / 3

Volumes ratio = ( 2 / 3 )³

Answer = 8 / 27

_________

Option ( d )

Answer 2

$\textbf {Hey there, here is the solution.}$

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Formula:

$\bigg( \dfrac{\text{Length 1}}{\text{Length 2}} \bigg)^2 = \dfrac{\text{Area 1}}{\text{Area 2}}$

.

STEP 1: Find the ratio of the length:

Plug in the known value:

$\bigg( \dfrac{\text{Length 1}}{\text{Length 2}} \bigg)^2 = \dfrac{4}{9}$

Evaluate the RHS:

$\dfrac{\text{Length 1}}{\text{Length 2}} = \dfrac{\sqrt{4}}{\sqrt{9}}$

$\dfrac{\text{Length 1}}{\text{Length 2}} = \dfrac{2}{3}$

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STEP 2: Find the ratio of the volume:

Formula:

$\bigg( \dfrac{\text{Length 1}}{\text{Length 2}} \bigg)^3 = \dfrac{\text{Volume 1}}{\text{Volume 2}}$

Plug in the known value:

$\dfrac{\text{Volume 1}}{\text{Volume 2}} = \bigg( \dfrac{2}{3} \bigg)^3$

Evaluate the RHS:

$\dfrac{\text{Volume 1}}{\text{Volume 2}} = \dfrac{2^3}{3^3}$

$\dfrac{\text{Volume 1}}{\text{Volume 2}} = \dfrac{8}{27}$

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Answer: (d) 8 : 27

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$\textbf {Cheers.}$