Let's solve the given problem step by step.
### Step 1: Relationship Between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]
We know that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex]. This means that [tex]\( y \)[/tex] is equal to some constant [tex]\( k \)[/tex] multiplied by [tex]\( x \)[/tex]:
[tex]\[
y = kx
\][/tex]
### Step 2: Finding the Constant [tex]\( k \)[/tex]
We are given that [tex]\( y = 39 \)[/tex] when [tex]\( x = 6 \)[/tex]. We can use this information to find the constant of proportionality [tex]\( k \)[/tex].
Substitute [tex]\( y = 39 \)[/tex] and [tex]\( x = 6 \)[/tex] into the equation:
[tex]\[
39 = k \cdot 6
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{39}{6} = 6.5
\][/tex]
### Step 3: Equation that Relates [tex]\( y \)[/tex] and [tex]\( x \)[/tex]
Now that we have found the constant [tex]\( k \)[/tex], we can write the equation that relates [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = 6.5x
\][/tex]
### Step 4: Finding [tex]\( y \)[/tex] when [tex]\( x = \frac{13}{2} \)[/tex]
Next, we need to find the value of [tex]\( y \)[/tex] when [tex]\( x = \frac{13}{2} \)[/tex].
Substitute [tex]\( x = \frac{13}{2} \)[/tex] into the equation:
[tex]\[
y = 6.5 \cdot \frac{13}{2}
\][/tex]
Calculating the result:
[tex]\[
y = 6.5 \cdot 6.5 = 42.25
\][/tex]
So, [tex]\( y = 42.25 \)[/tex] when [tex]\( x = \frac{13}{2} \)[/tex].
### Step 5: Finding [tex]\( y \)[/tex] when [tex]\( x = 29 \)[/tex]
We are also asked to find the value of [tex]\( y \)[/tex] when [tex]\( x = 29 \)[/tex].
Substitute [tex]\( x = 29 \)[/tex] into the equation:
[tex]\[
y = 6.5 \cdot 29
\][/tex]
Calculating the result:
[tex]\[
y = 188.5
\][/tex]
So, [tex]\( y = 188.5 \)[/tex] when [tex]\( x = 29 \)[/tex].
### Summary
- The equation that relates [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is [tex]\( y = 6.5x \)[/tex].
- When [tex]\( x = \frac{13}{2} \)[/tex], [tex]\( y = 42.25 \)[/tex].
- When [tex]\( x = 29 \)[/tex], [tex]\( y = 188.5 \)[/tex].
