Find the mathematical model for the following statement. Remember to use k as the constant of proportionality.

z varies jointly as the 4th power of x and the 5th power of y.



To form the mathematical model, we use the fact that **joint variation** means that one quantity (here, \( z \)) varies with the product of two or more other quantities (here, \( x \) and \( y \)).

The statement says that **\( z \) varies jointly** as the **4th power of \( x \)** and the **5th power of \( y \)**. This translates into the following equation:

\[
z = k \cdot x^4 \cdot y^5
\]

Where:
- \( z \) is the dependent variable,
- \( x \) and \( y \) are the independent variables,
- \( k \) is the constant of proportionality.

This is the mathematical model for the given variation statement.

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the equation change if \( z \) varied inversely with \( y \)?
2. What happens to \( z \) if \( x \) doubles and \( y \) remains the same?
3. How does the constant \( k \) affect the magnitude of \( z \)?
4. Can you graph this relationship if \( k = 1 \), \( x = 1 \), and \( y \) ranges from 1 to 10?
5. How does the equation change if \( z \) varies directly with \( x^4 \), but inversely with \( y^5 \)?

**Tip**: When dealing with joint variation, always pay attention to the powers of the variables, as they dictate the strength of the relationship.

Find the mathematical model for the following statement. Remember to use k as the constant of proportionality. z varies jointly as the 4th power of x and the 5th power of y.

Explore how to find the mathematical model for the joint variation of z with the 4th power of x and the 5th power of y. This problem introduces the concept of joint variation and its algebraic representation using exponents. Suitable for high school students in Grades 9-12.

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