For the​ statement, find the constant of variation and the variation equation. 
y varies jointly as x and the cube of​ z; yequals405 when xequals3 and zequals3
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Part 1
Find the constant of variation k.
kequals
  
enter your response here ​(Simplify your​ answer.)


The statement says that \( y \) varies jointly as \( x \) and the cube of \( z \). The general form for this type of variation is:

\[
y = kxz^3
\]

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

We are given that:
- \( y = 405 \)
- \( x = 3 \)
- \( z = 3 \)

Substitute these values into the equation to solve for \( k \):

\[
405 = k(3)(3^3)
\]

Since \( 3^3 = 27 \), the equation becomes:

\[
405 = k(3)(27)
\]

\[
405 = 81k
\]

Now, solve for \( k \):

\[
k = \frac{405}{81} = 5
\]

Thus, the constant of variation \( k \) is:

\[
k = 5
\]

### Part 2: The variation equation

Now that we know \( k = 5 \), substitute it back into the variation equation:

\[
y = 5xz^3
\]

So the variation equation is:

\[
y = 5xz^3
\]

Would you like further details or have any questions about this? 

Here are 5 related questions to practice:

1. If \( y \) varies jointly as \( x \) and the square of \( z \), how would the equation change?
2. What happens to \( y \) if \( z \) is doubled while \( x \) stays constant?
3. Solve for \( y \) when \( x = 2 \), \( z = 4 \), and \( k = 5 \).
4. How do you identify when a problem involves joint variation instead of direct or inverse variation?
5. Can you rewrite the equation if \( y \) varies jointly as \( x \), \( z^3 \), and inversely as \( w \)?

**Tip:** Joint variation problems involve multiple variables, where one variable depends on the product (or quotient) of others.

For the statement, find the constant of variation and the variation equation. y varies jointly as x and the cube of z; y equals 405 when x equals 3 and z equals 3. Find the constant of variation k.

Learn how to solve a joint variation problem where y varies jointly as x and the cube of z. The problem gives y = 405 when x = 3 and z = 3. The solution shows how to find the constant of variation k and the variation equation. Suitable for students in Grades 9-11.

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