Let's solve the problem step by step.

Part (a): Writing a relationship between $x$ and $y$ using the symbol $\propto$:

The relationship $y \propto x^3$ seems likely because the values of $y$ grow rapidly as $x$ increases, suggesting cubic proportionality.

Verifying:

1. When $x = 2$, $y = 24$. Check if $y = kx^3$: $24 = k(2^3) \implies k = \frac{24}{8} = 3$

2. Check for $x = 3$, $y = 81$: $y = kx^3 \implies y = 3(3^3) = 3(27) = 81$

3. Check for $x = 4$, $y = 192$: $y = 3(4^3) = 3(64) = 192$

4. Check for $x = 5$, $y = 375$: $y = 3(5^3) = 3(125) = 375$

Thus, the relationship is: $y \propto x^3$

Part (b): Writing an equation involving $x$ and $y$:

From the proportional relationship $y \propto x^3$, we include the constant of proportionality $k = 3$. The equation is: $y = 3x^3$

Part (c): Finding $x$ when $y = 3993$:

Using the equation $y = 3x^3$, solve for $x$: $3993 = 3x^3 \implies x^3 = \frac{3993}{3} = 1331$ $x = \sqrt[3]{1331} = 11$

Thus, when $y = 3993$, $x = 11$.

Final Answers:

(a) $y \propto x^3$
(b) $y = 3x^3$
(c) $x = 11$ when $y = 3993$.

Would you like a detailed breakdown of the calculations? Let me know!

Here are 5 related questions for practice:

1. If $y \propto x^4$ and $y = 256$ when $x = 4$, find $k$ and the equation.
2. Solve for $x$ when $y = 500$ in the equation $y = 5x^3$.
3. If $y = 2x^3 + 5$, find $y$ when $x = 6$.
4. Derive the proportionality constant if $y \propto x^2$ and $y = 50$ when $x = 5$.
5. Explain how cubic relationships differ from linear relationships in terms of growth rate.

Tip: When dealing with proportional relationships, always test for the constant $k$ to confirm the equation!