To solve for $z$ when $y = 14$ and $x = 7$, given the variation equation:

$z = 4.7 \cdot x^3 \cdot y^2$

we can substitute the values of $x$ and $y$ into the equation as follows:

1. Substitute $x = 7$ and $y = 14$ into the equation:

   $z = 4.7 \cdot (7)^3 \cdot (14)^2$

2. Calculate each component step-by-step:

   • $7^3 = 343$
   • $14^2 = 196$

3. Substitute these values back:

   $z = 4.7 \cdot 343 \cdot 196$

4. Multiply the values:

   $z = 4.7 \cdot 67228$

5. Finally, multiply by $4.7$:

   $z = 316971.6$

Therefore, the answer is:

$z = 316971.6$

Would you like further details on any part of this calculation?

Related Questions:

1. How does joint variation differ from direct variation?
2. How do you derive the constant of proportionality?
3. What are some examples of real-world applications of joint variation?
4. How would the equation change if $z$ varied with the fourth power of $x$?
5. What steps would you take if asked to find $x$ or $y$ instead of $z$?

Tip: Always double-check your substitutions and calculations, especially when dealing with exponents, as small errors can lead to large discrepancies in answers.