To solve the problem, we need to find the constant of variation for the relation and use it to write an equation, then determine the value of [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] and [tex]\( z = 4 \)[/tex].
1. Understand the relationship:
The problem states that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex] and [tex]\( z \)[/tex]. This means we can express [tex]\( y \)[/tex] as:
[tex]\[
y = kxz
\][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
2. Given values to find [tex]\( k \)[/tex]:
We are given:
[tex]\[
y = 4 \quad \text{when} \quad x = 6 \quad \text{and} \quad z = 1
\][/tex]
Substitute these values into the equation to find [tex]\( k \)[/tex]:
[tex]\[
4 = k \cdot 6 \cdot 1
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{4}{6} = \frac{2}{3}
\][/tex]
3. Write the equation using [tex]\( k \)[/tex]:
Now that we have [tex]\( k \)[/tex], the equation becomes:
[tex]\[
y = \frac{2}{3} x z
\][/tex]
4. Solve for [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] and [tex]\( z = 4 \)[/tex]:
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( z = 4 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3} \cdot 7 \cdot 4
\][/tex]
Simplify the expression:
[tex]\[
y = \frac{2}{3} \cdot 28 = \frac{56}{3}
\][/tex]
Therefore, the correct constant of variation is [tex]\(\frac{2}{3}\)[/tex] and the value of [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] and [tex]\( z = 4 \)[/tex] is [tex]\(\frac{56}{3}\)[/tex].
So the correct answer is:
[tex]\[
\text{d. } y = \frac{2}{3} x z; \quad y(7, 4) = \frac{56}{3}
\][/tex]
