If \frac{2a}{3}=\frac{4b}{5}=\frac{3c}{4} then what is the value of \frac{18}{a}\sqrt{a^{2}+c^{2}-b^{2}}?

Explanation

Let \frac{2a}{3}=\frac{4b}{5}=\frac{3c}{4} = k. Then a = \frac{3k}{2}, b = \frac{5k}{4}, c = \frac{4k}{3}. Substituting into the expression: \frac{18}{3k/2} \sqrt{\frac{9k^2}{4} + \frac{16k^2}{9} - \frac{25k^2}{16}} = \frac{12}{k} \cdot k \sqrt{\frac{1296 + 1024 - 900}{576}} = 12 \sqrt{\frac{1420}{576}} = \frac{12}{24} \sqrt{1420} = \frac{1}{2} \sqrt{355 \times 4} = \sqrt{355}.

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