To solve this problem, we need to understand the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] since [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex]. This means that [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
1. Determine the constant [tex]\( k \)[/tex]:
We are given that [tex]\( y = 48 \)[/tex] when [tex]\( x = 6 \)[/tex]. We can use this information to find the constant [tex]\( k \)[/tex]:
[tex]\[
y = kx \implies 48 = k(6)
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{48}{6} = 8
\][/tex]
2. Form the direct variation equation:
Now that we have [tex]\( k \)[/tex], we can write the equation that describes the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = kx \implies y = 8x
\][/tex]
3. Use the derived equation:
We need to find which expression can be used to find [tex]\( y \)[/tex] for a given [tex]\( x \)[/tex]. The derived equation [tex]\( y = 8x \)[/tex] must hold true.
4. Verify each option:
- Option 1: [tex]\( y = \frac{48}{6}(2) \)[/tex]
[tex]\[
y = 8 \times 2 = 16
\][/tex]
This is consistent with our derived equation [tex]\( y = 8x \)[/tex] when [tex]\( x = 2 \)[/tex]. Thus, this is a correct expression.
- Option 2: [tex]\( y = \frac{6}{48}(2) \)[/tex]
[tex]\[
y = \frac{6}{48} \times 2 = \frac{1}{4} \times 2 = \frac{2}{4} = \frac{1}{2}
\][/tex]
This does not match our derived equation [tex]\( y = 8x \)[/tex]. Therefore, this option is incorrect.
- Option 3: [tex]\( y = \frac{(4 g)(6)}{2} \)[/tex]
This option seems not to make sense dimensionally. The term [tex]\( (4 g) \)[/tex] is unclear, and assuming it implies a numerical value, it still doesn't align with the required equation format.
- Option 4: [tex]\( y = \frac{2}{(48)(6)} \)[/tex]
[tex]\[
y = \frac{2}{48 \times 6} = \frac{2}{288} = \frac{1}{144}
\][/tex]
This does not match our derived equation [tex]\( y = 8x \)[/tex]. Therefore, this option is incorrect.
After carefully examining these options, we find that the correct expression to use is:
[tex]\[
y = \frac{48}{6}(2)
\][/tex]
