If $ 3.2 : x :: x : 16.2 $, and $ x 0 $, then find the value of $ x $.

Question

If $ 3.2 : x :: x : 16.2 $, and $ x > 0 $, then find the value of $ x $.

Examoogle · Accepted Answer

The correct answer is Option B: 7.2. In a continuous proportion $a : x :: x : b$, the middle term is the mean proportional, given by $x = \sqrt{ab}$. Calculate $\sqrt{3.2 	imes 16.2} = \sqrt{\frac{32}{10} 	imes \frac{162}{10}} = \sqrt{\frac{5184}{100}}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: In a continuous proportion $a : x :: x : b$, the middle term is the mean proportional, given by $x = \sqrt{ab}$. Calculate $\sqrt{3.2 	imes 16.2} = \sqrt{\frac{32}{10} 	imes \frac{162}{10}} = \sqrt{\frac{5184}{100}}$.

Formula Used: $$x^2 = a 	imes b$$

Given: Given the proportion 3.2 : x :: x : 16.2, where x > 0.

Steps: Setup the equation: A proportion $a : x :: x : b$ can be written as $\frac{a}{x} = \frac{x}{b}$, which implies $x^2 = ab$. Here $a = 3.2$ and $b = 16.2$.. Perform multiplication: Multiply the two numbers: $3.2 	imes 16.2 = 51.84$.. Calculate the square root: Take the square root of $51.84$ to find $x$. Since $x > 0$, we only consider the positive root.

Answer: B is correct because the mean proportional $x = \sqrt{3.2 	imes 16.2} = \sqrt{51.84} = 7.2$.

Exam Tip: This concept is a subset of 'Continued Proportion' which appears frequently in Speed-Distance-Time problems involving geometric ratios or areas.

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