Question

Suppose that z varies jointly with x and y.

Find the constant of proportionality k if z = 652.8 when y = 12 and x = 8.
k =
Using the k from above write the variation equation in terms of x and y.
z =
Using the k from above find z given that y = 34 and x = 22.
z =

If needed, round answer to 3 decimal places.
Enter DNE for Does Not Exist, oo for Infinity

Answer 1

Since \( z \) varies jointly with \( x \) and \( y \), we can express this relationship with the equation:

\[
z = kxy
\]

where \( k \) is the constant of proportionality.

### Step 1: Find the constant of proportionality \( k \)

We know that \( z = 652.8 \) when \( y = 12 \) and \( x = 8 \). We can plug these values into the equation to solve for \( k \):

\[
652.8 = k \cdot 8 \cdot 12
\]

Calculating the right side:

\[
652.8 = k \cdot 96
\]

To find \( k \), divide both sides by 96:

\[
k = \frac{652.8}{96}
\]

Calculating \( k \):

\[
k = 6.8
\]

### Step 2: Write the variation equation in terms of \( x \) and \( y \)

Now that we have found \( k \), we can write the variation equation as:

\[
z = 6.8xy
\]

### Step 3: Find \( z \) when \( y = 34 \) and \( x = 22 \)

Now we can use the equation to find \( z \) when \( y = 34 \) and \( x = 22 \):

\[
z = 6.8 \cdot 22 \cdot 34
\]

Calculating the right side:

\[
z = 6.8 \cdot 748
\]

Calculating \( 6.8 \cdot 748 \):

\[
z = 5084.8
\]

### Summary of Results

- Constant of proportionality \( k = 6.8 \)
- Variation equation in terms of \( x \) and \( y \): \( z = 6.8xy \)
- Value of \( z \) when \( y = 34 \) and \( x = 22 \): \( z = 5084.8 \)

So, the answers are:
- \( k = 6.8 \)
- \( z = 6.8xy \)
- \( z = 5084.8 \) (Rounded to 3 decimal places if applicable; here it is already rounded.)