Question 22

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R

Assertion A : If the volumes of two cubes are in the ratio of 3 : 2 then their surface areas are in the ratio of 4 : 9.

Reason R : If the surface areas of two cubes are in the ratio $$S_{1} : S_{2}$$, then their volumes are in the ratio $$S_{1}^{3/2} : S_{2}^{3/2}$$

In the light of the above statements, choose the correct answer from the options given below :

Solution

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  • Assertion A is false: If volumes are in the ratio $$3:2$$, then the side ratio is the cube root, $$\sqrt[3]{3}:\sqrt[3]{2}$$. Squaring this gives the surface area ratio as $$3^{\frac{2}{3}}:2^{\frac{2}{3}}$$.
  • Reason R is true: If surface areas are $$S_1 : S_2$$, the side ratio is $$\sqrt{S_1} : \sqrt{S_2}$$ (or $$S_1^{1/2} : S_2^{1/2}$$). Since volume is the cube of the side, the volume ratio is $$(S_1^{1/2})^3 : (S_2^{1/2})^3$$, which simplifies to $$S_1^{3/2} : S_2^{3/2}$$. 

    • Thus, the correct answer is Option D.

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