To solve this problem, we need to follow these steps systematically:
1. Understand the given relationship:
We know that [tex]\(\sqrt{q}\)[/tex] is inversely proportional to [tex]\(r\)[/tex]. This tells us that [tex]\(\sqrt{q} = \frac{k}{r}\)[/tex], where [tex]\(k\)[/tex] is a constant of proportionality.
2. Use the initial given values to find [tex]\(k\)[/tex]:
We are told that [tex]\(\sqrt{q} = 9\)[/tex] when [tex]\(r = 4\)[/tex]. Substituting these values into the equation, we get:
[tex]\[
9 = \frac{k}{4}
\][/tex]
To solve for [tex]\(k\)[/tex], we multiply both sides of the equation by 4:
[tex]\[
k = 9 \times 4 = 36
\][/tex]
3. Determine [tex]\(r\)[/tex] when [tex]\(q=4\)[/tex]:
First, find [tex]\(\sqrt{q}\)[/tex] for the new value of [tex]\(q\)[/tex]:
[tex]\[
\sqrt{q} = \sqrt{4} = 2
\][/tex]
4. Use the constant [tex]\(k\)[/tex] to find the new [tex]\(r\)[/tex]:
We now use the relationship [tex]\(\sqrt{q} = \frac{k}{r}\)[/tex]. Plugging in the value of [tex]\(\sqrt{q} = 2\)[/tex] and [tex]\(k = 36\)[/tex], we get:
[tex]\[
2 = \frac{36}{r}
\][/tex]
Solving for [tex]\(r\)[/tex], we multiply both sides of the equation by [tex]\(r\)[/tex]:
[tex]\[
2r = 36
\][/tex]
Dividing both sides by 2, we get:
[tex]\[
r = \frac{36}{2} = 18
\][/tex]
Therefore, the value of [tex]\(r\)[/tex] when [tex]\(q = 4\)[/tex] is [tex]\(18\)[/tex].
