Let's break down the problem into a set of clear, logical steps. We are given a relationship where [tex]\( E \)[/tex] varies directly as [tex]\( F \)[/tex] and inversely as the cube root of [tex]\( G \)[/tex]. Mathematically, this is expressed as:
[tex]\[ E = k \frac{F}{\sqrt[3]{G}} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
Given values from the first set are:
[tex]\[ E_1 = 6 \][/tex]
[tex]\[ F_1 = 0.4 \][/tex]
[tex]\[ G_1 = 0.008 \][/tex]
First, we need to determine the constant [tex]\( k \)[/tex] using these values. Plugging them into the equation:
[tex]\[ 6 = k \frac{0.4}{\sqrt[3]{0.008}} \][/tex]
Knowing that [tex]\( \sqrt[3]{0.008} = 0.2 \)[/tex]:
[tex]\[ 6 = k \frac{0.4}{0.2} \][/tex]
[tex]\[ 6 = k \times 2 \][/tex]
[tex]\[ k = \frac{6}{2} \][/tex]
[tex]\[ k = 3 \][/tex]
Now that we have [tex]\( k \)[/tex], we use this constant with the second set of values to find [tex]\( P \)[/tex]. The second set gives us:
[tex]\[ E_2 = 3.5 \][/tex]
[tex]\[ F_2 = 7 \][/tex]
[tex]\[ G_2 = P \][/tex]
Plugging these values into the relationship:
[tex]\[ 3.5 = 3 \frac{7}{\sqrt[3]{P}} \][/tex]
[tex]\[ \sqrt[3]{P} = 3 \frac{7}{3.5} \][/tex]
[tex]\[ \sqrt[3]{P} = 6 \][/tex]
Cubing both sides to solve for [tex]\( P \)[/tex]:
[tex]\[ P = 6^3 \][/tex]
[tex]\[ P = 216 \][/tex]
Thus, the value of [tex]\( P \)[/tex] is:
[tex]\[ \boxed{216} \][/tex]
Therefore, the correct answer is B. 216.
