Since $y$ is directly proportional to $x^3$, we can express this relationship as:

$y = kx^3$

where $k$ is the constant of proportionality.

Step 1: Find the constant $k$

We are given that $y = 36$ when $x = 8$. Substitute these values into the equation:

$36 = k(8^3)$ $36 = k(512)$ $k = \frac{36}{512}$ $k = \frac{9}{128}$

Step 2: Use the constant to find $y$ when $x = 7$

Now, substitute $x = 7$ and $k = \frac{9}{128}$ into the equation $y = kx^3$:

$y = \frac{9}{128}(7^3)$ $y = \frac{9}{128}(343)$ $y = \frac{9 \times 343}{128}$ $y = \frac{3087}{128}$ $y \approx 24.12$

Thus, when $x = 7$, $y \approx 24.12$.

Would you like more details on any step, or do you have any questions?

Here are five related questions:

1. How would you solve if $y$ were inversely proportional to $x^3$?
2. What is the difference between direct and inverse proportionality?
3. How can we verify if a proportional relationship holds between two variables?
4. What happens to $y$ as $x$ increases in a direct proportionality to $x^3$?
5. Can proportional relationships be nonlinear?

Tip: Always substitute known values into a proportionality equation first to find the constant.