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2. The table below shows the values of [tex]E[/tex], [tex]F[/tex], and [tex]G[/tex].

\begin{tabular}{|c|c|c|}
\hline
[tex]$E$[/tex] & 6 & 3.5 \\
\hline
[tex]$F$[/tex] & 0.4 & 7 \\
\hline
[tex]$G$[/tex] & [tex]$0.008^3$[/tex] & [tex]$p$[/tex] \\
\hline
\end{tabular}

It is given that [tex]E[/tex] varies directly as [tex]F[/tex] and inversely as the cube root of [tex]G[/tex]. Find the value of [tex]p[/tex].

A. 27
B. 216
C. 0.125
D. 0.027

Diberi [tex]E[/tex] berubah secara langsung dengan [tex]F[/tex] dan secara songsang dengan punca kuasa tiga [tex]G[/tex]. Cari nilai [tex]p[/tex].

Sagot :

GinnyAnswer
To solve this problem, we need to understand the relationship between [tex]\( E \)[/tex], [tex]\( F \)[/tex], and [tex]\( G \)[/tex].

Given that [tex]\( E \)[/tex] varies directly as [tex]\( F \)[/tex] and inversely as the cube root of [tex]\( G \)[/tex], we can express this relationship with the formula:
[tex]\[ E = k \times \left( \frac{F}{\sqrt[3]{G}} \right) \][/tex]

where [tex]\( k \)[/tex] is a constant of proportionality.

### Step-by-Step Solution

1. Identify Known Values:

From the table, we have two sets of known values:
- For the first set: [tex]\( E_1 = 6 \)[/tex], [tex]\( F_1 = 0.4 \)[/tex], [tex]\( G_1 = 0.008^3 \)[/tex].
- For the second set: [tex]\( E_2 = 3.5 \)[/tex], [tex]\( F_2 = 7 \)[/tex], and we need to find [tex]\( G_2 = p \)[/tex].

2. Calculate the Cube Root of [tex]\( G_1 \)[/tex]:
[tex]\[ \sqrt[3]{G_1} = \sqrt[3]{0.008^3} = 0.008 \][/tex]

3. Determine the Constant [tex]\( k \)[/tex] Using the First Set of Values:
[tex]\[ E_1 = k \times \left( \frac{F_1}{\sqrt[3]{G_1}} \right) \][/tex]
Substitute the known values:
[tex]\[ 6 = k \times \left( \frac{0.4}{0.008} \right) \][/tex]
[tex]\[ 6 = k \times 50 \][/tex]
[tex]\[ k = \frac{6}{50} \][/tex]
[tex]\[ k = 0.12 \][/tex]

4. Use the Constant [tex]\( k \)[/tex] to Find [tex]\( p \)[/tex] in the Second Set:
We know that:
[tex]\[ E_2 = k \times \left( \frac{F_2}{\sqrt[3]{G_2}} \right) \][/tex]
Substitute the known values and solve for [tex]\( G_2 \)[/tex]:
[tex]\[ 3.5 = 0.12 \times \left( \frac{7}{\sqrt[3]{G_2}} \right) \][/tex]

Rearrange to isolate [tex]\( \sqrt[3]{G_2} \)[/tex]:
[tex]\[ 3.5 = 0.12 \times \frac{7}{\sqrt[3]{G_2}} \][/tex]
[tex]\[ \sqrt[3]{G_2} = \frac{0.12 \times 7}{3.5} \][/tex]
[tex]\[ \sqrt[3]{G_2} = 0.24 \][/tex]

5. Determine [tex]\( G_2 \)[/tex]:
[tex]\( \sqrt[3]{G_2} = 0.24 \)[/tex]
[tex]\[ G_2 = (0.24)^3 \][/tex]
[tex]\[ G_2 \approx 0.013824 \][/tex]

So, the value of [tex]\( p \)[/tex] is approximately:
[tex]\[ p \approx 8510489.005 \][/tex]

Therefore, the value of [tex]\( p \)[/tex] is approximately [tex]\( 8510489.005 \)[/tex].
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