To solve the problem, we need to use the relationship that [tex]\(E\)[/tex] varies directly as [tex]\(F\)[/tex] and inversely as the cube root of [tex]\(G\)[/tex]. The mathematical representation of this relationship can be written as:
[tex]\[ E = k \cdot \frac{F}{\sqrt[3]{G}} \][/tex]
where [tex]\(k\)[/tex] is the constant of variation.
First, we need to determine the value of [tex]\(k\)[/tex] using the first set of given values from the table:
[tex]\[ E = 6 \][/tex]
[tex]\[ F = 0.4 \][/tex]
[tex]\[ G = 0.008 \][/tex]
Substituting these values into the equation, we get:
[tex]\[ 6 = k \cdot \frac{0.4}{\sqrt[3]{0.008}} \][/tex]
The cube root of [tex]\(0.008\)[/tex] is [tex]\(0.2\)[/tex], so the equation becomes:
[tex]\[ 6 = k \cdot \frac{0.4}{0.2} \][/tex]
[tex]\[ 6 = k \cdot 2 \][/tex]
Solving for [tex]\(k\)[/tex], we find:
[tex]\[ k = \frac{6}{2} = 3 \][/tex]
Now that we have the value of [tex]\(k\)[/tex], we can use the second set of values to find [tex]\(P\)[/tex], where [tex]\(G = P\)[/tex]:
[tex]\[ E = 8.5 \][/tex]
[tex]\[ F = 7 \][/tex]
Substituting these values into the equation with the known constant [tex]\(k = 3\)[/tex], we get:
[tex]\[ 8.5 = 3 \cdot \frac{7}{\sqrt[3]{P}} \][/tex]
Solving for [tex]\(\sqrt[3]{P}\)[/tex]:
[tex]\[ 8.5 = 21 \div \sqrt[3]{P} \][/tex]
[tex]\[ \sqrt[3]{P} = \frac{21}{8.5} \][/tex]
[tex]\[ \sqrt[3]{P} \approx 2.470588 \][/tex]
Now, to find [tex]\(P\)[/tex], we cube both sides:
[tex]\[ P = (2.470588)^3 \][/tex]
[tex]\[ P \approx 15.07999185833504 \][/tex]
Thus, the value of [tex]\(P\)[/tex] is approximately [tex]\(15.08\)[/tex].
