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Mathematics

Find the mean proportional between 6 + 33 and 8−433\sqrt{3} \text{ and } 8 - 4\sqrt{3}

Ratio Proportion

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Answer

Let mean proportional between 6 + 33 and 8−433\sqrt{3} \text{ and } 8 - 4\sqrt{3} be x

∴6+33:x=x:8−43⇒6+33x=x8−43⇒x2=(6+33)(8−43)⇒x2=48−243+243−36⇒x2=48−36⇒x2=12⇒x=23.\therefore 6 + 3\sqrt{3} : x = x : 8 - 4\sqrt{3} \\[1em] \Rightarrow \dfrac{6 + 3\sqrt{3}}{x} = \dfrac{x}{8 - 4\sqrt{3}} \\[1em] \Rightarrow x^2 = (6 + 3\sqrt{3})(8 - 4\sqrt{3}) \\[1em] \Rightarrow x^2 = 48 - 24\sqrt{3} + 24\sqrt{3} - 36 \\[1em] \Rightarrow x^2 = 48 - 36 \\[1em] \Rightarrow x^2 = 12 \\[1em] \Rightarrow x = 2\sqrt{3}.

Hence, x = 232\sqrt{3}.

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