The value of $m$ varies inversely as the square of $n$. When $n=3$, $m=6$. What is the positive value of $n$ when $m=13.5$?
The problem states that \(m\) varies inversely as the square of \(n\). This relationship can be expressed by the equation:
\[m = \frac{k}{n^2}\]
where \(k\) is a non-zero constant. We are given an initial condition where when \(n = 3\), \(m = 6\). We can use this information to determine the value of the constant \(k\). Substituting these values into the equation, we have:
\[6 = \frac{k}{3^2}\]
\[6 = \frac{k}{9}\]
Multiplying both sides by 9 gives:
\[k = 6 \times 9 = 54\]
Thus, the relationship between \(m\) and \(n\) is given by the equation:
\[m = \frac{54}{n^2}\]
Now that we have the specific equation relating \(m\) and \(n\), we can solve for the value of \(n\) when \(m = 13.5\). Setting \(m\) to 13.5, we get:
\[13.5 = \frac{54}{n^2}\]
To solve for \(n^2\), we rearrange the equation:
\[n^2 = \frac{54}{13.5}\]
Performing the division:
\[n^2 = 4\]
Taking the square root of both sides to find \(n\):
\[n = \pm \sqrt{4} = \pm 2\]
Since the question asks for the positive value of \(n\), we conclude that \(n = 2\). Comparing this with the provided choices (9, 2, 18, 4), the correct answer is 2.
In physics, the intensity of light \(I\) from a point source varies inversely as the square of the distance \(d\) from the source (\(I = \frac{k}{d^2}\)). If you know the light intensity at a certain distance, you can use the same inverse square relationship to calculate the distance required for a desired intensity level, helping in lighting design and photography.