To determine the mean proportional between two numbers, we need to find the square root of the product of those two numbers. Given the numbers are \(8 + 3\sqrt{3}\) and \(16 - 6\sqrt{3}\), we will find the product of these numbers and then its square root.
The formula for the mean proportional \(x\) between two numbers \(a\) and \(b\) is:
\(x = \sqrt{a \cdot b}\)
Substituting the given numbers:
\(a = 8 + 3\sqrt{3}\) and \(b = 16 - 6\sqrt{3}\)
First, we calculate the product:
\(\begin{align*} a \cdot b & = (8 + 3\sqrt{3})(16 - 6\sqrt{3}) \\ & = 8 \cdot 16 + 8 \cdot (-6\sqrt{3}) + 3\sqrt{3} \cdot 16 + 3\sqrt{3} \cdot (-6\sqrt{3}) \\ & = 128 - 48\sqrt{3} + 48\sqrt{3} - 54 \\ & = 128 - 54 \\ & = 74 \end{align*}\)
Now, the mean proportional is:
\(x = \sqrt{74}\)
Thus, the correct answer is: \(\sqrt{74}\)
This matches the correct answer option provided: \(\sqrt{74}\).
The other options given do not equal to the product of the numbers, hence they are incorrect.
Similar Questions
If \(x\) is the mean proportional between \((a+b)\) and \((a-b)\), then find the value of \(x\).
In a fitness park, a straight walking path is divided into three consecutive segments. The lengths of the first and third segments are 18 m and 50 m respectively. If the length of the middle segment is the mean proportional between the other two, what is its length?
If the height at which a ladder touches the wall is the mean proportional between 6 meters and 24 meters, at what height does the ladder touch the wall?
A rectangular garden has a length of 16 meters and a width of x meters. If the width is the mean proportional between 4 meters and 64 meters, what is the value of x?
Find the mean proportional between \((2+\sqrt{3})\) and \((10-\sqrt{75})\).
The mean proportional between two positive numbers is 30. If the difference between the two numbers is 45, what is the smaller of the two numbers?
In all, how many kids did not want to see the galaxy?
In all, how many kids wanted to see the moon?
How many kids wanted to see at least 2 objects?
How many children like both brown and white but not black?
Important Questions from Mean Proportional
- Find the mean proportional between 19 and 76.
- Find the mean proportional of 324 and 16.
- Find the mean proportional between 28 and 252.
- Find the mean proportional of 400 and 4.
- Find the mean proportional of 225 and 4.
