Question

suppose that z varies jointly with x and y. find the constant of proportionality k if z = 472 when y = 10 and x = 8. k = \boxed{} using the k from above write the variation equation in terms of x and y. z = \boxed{} using the k from above find z given that y = 18 and x = 22. z = \boxed{} if needed, round answer to 3 decimal places. enter dne for does not exist, oo for infinity

Explanation:

Step1: Recall joint variation formula

If \( z \) varies jointly with \( x \) and \( y \), the formula is \( z = kxy \), where \( k \) is the constant of proportionality.

Step2: Solve for \( k \)

We know \( z = 472 \), \( x = 8 \), and \( y = 10 \). Substitute these values into the formula:
\( 472 = k \times 8 \times 10 \)
Simplify the right - hand side: \( 8\times10 = 80 \), so the equation becomes \( 472=80k \).
To solve for \( k \), divide both sides by 80: \( k=\frac{472}{80}=5.9 \)

Step3: Write the variation equation

Now that we have \( k = 5.9 \), substitute \( k \) back into the joint variation formula \( z = kxy \). So the variation equation is \( z = 5.9xy \)

Step4: Find \( z \) when \( x = 22 \) and \( y = 18 \)

Use the variation equation \( z = 5.9xy \). Substitute \( x = 22 \) and \( y = 18 \):
\( z=5.9\times22\times18 \)
First, calculate \( 22\times18 = 396 \). Then, multiply by 5.9: \( z = 5.9\times396=2336.4 \)

Answer:

• For \( k \): \( 5.9 \)
• For the variation equation: \( 5.9xy \)
• For \( z \) when \( x = 22 \) and \( y = 18 \): \( 2336.4 \)