Question : The lengths, the breadths, and the volumes of two cuboids are in the ratios of 4 : 5, 3 : 4, and 2 : 3, respectively. What is the ratio of their heights?
Option 1: 3 : 5
Option 2: 10 : 9
Option 3: 8 : 15
Option 4: 2 : 3
Correct Answer: 10 : 9
Solution :
The volume of a cuboid is $V = lbh$, where l, b and h are the length, breadth and height respectively.
Let the volume of the first cuboid be $V_1 = l_1b_1h_1$
Let the volume of the second cuboid be $V_2 = l_2b_2h_2$
On dividing the two equations and substituting the given ratios,
⇒ $\frac{V_1}{V_2} = \frac{l_1b_1h_1}{l_2b_2h_2}$
⇒ $\frac{2}{3} = \frac{4\times3\times h_1}{5\times4\times h_2}$
⇒ $\frac{h_1}{h_2} = \frac{2\times4\times 5}{4\times3\times 3}= \frac{10}{9}$
$\therefore$ The ratio of the heights of the two cuboids is 10 : 9.
Hence, the correct answer is 10 : 9.
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