Sure, let's solve this step by step!
Given that [tex]\(P\)[/tex] varies inversely as [tex]\(V\)[/tex], we can express this as:
[tex]\[ P = \frac{k_1}{V} \][/tex]
where [tex]\(k_1\)[/tex] is a constant.
Additionally, [tex]\(V\)[/tex] varies directly as [tex]\(R^2\)[/tex], which gives us:
[tex]\[ V = k_2 R^2 \][/tex]
where [tex]\(k_2\)[/tex] is another constant.
To find the relationship between [tex]\(P\)[/tex] and [tex]\(R\)[/tex], we can combine these two relationships. Substituting [tex]\(V\)[/tex] from the second equation into the first equation, we get:
[tex]\[ P = \frac{k_1}{k_2 R^2} \][/tex]
This can be simplified to:
[tex]\[ P = \frac{k}{R^2} \][/tex]
where [tex]\(k = \frac{k_1}{k_2}\)[/tex] is a constant.
We are given the specific values [tex]\(P = 2\)[/tex] when [tex]\(R = 7\)[/tex]. We can use these to determine the constant [tex]\(k\)[/tex]:
[tex]\[ 2 = \frac{k}{7^2} \][/tex]
[tex]\[ 2 = \frac{k}{49} \][/tex]
[tex]\[ k = 2 \times 49 \][/tex]
[tex]\[ k = 98 \][/tex]
So, the relationship between [tex]\(P\)[/tex] and [tex]\(R\)[/tex] can be written as:
[tex]\[ P = \frac{98}{R^2} \][/tex]
Now, let's express this equation in the form given in the multiple choices. The equivalent form is:
[tex]\[ P R^2 = 98 \][/tex]
Thus, the correct relationship between [tex]\(P\)[/tex] and [tex]\(R\)[/tex] is:
[tex]\[ \boxed{P R^2 = 98} \][/tex]
Answer: B. [tex]\(P R^2 = 98\)[/tex].
