17. $x$ is directly proportional to the square of $y$ - Edexcel - GCSE Maths - Question 18 - 2021 - Paper 3

Given that $y$ is directly proportional to the cube of $z$, we can write:

$y = k_2 z^3$

Substituting this into the first equation gives:

$x = k_1 (k_2 z^3)^2 = k_1 k_2^2 z^6$

From the problem, when $x = 32$, $y = 2$. We can substitute these values to find the constants: Setting $y = 2$ gives us:

$2 = k_2 z^3 \Rightarrow z^3 = \frac{2}{k_2}$

Substituting this into the equation for $x$:

$32 = k_1 k_2^2 \left(\frac{2}{k_2}\right)^2 = k_1 k_2 \cdot 4$

Now, substituting values back in, we can express $x$ in terms of $z$:

From the previous step, we can derive:

Substituting for $k_1 k_2$, we can express:

$x = c z^6$

where $c$ is a constant that we can determine. Thus, the final formula for $x$ in terms of $z$ is:

$x = c z^6$