Let's solve this step by step:

Step 1: Find the constant of proportionality, $k$

We are given that $y = 58.8$ when $x = 7$. In a direct variation, the equation is:

$y = kx$

Substitute the values $y = 58.8$ and $x = 7$ into the equation:

$58.8 = k \times 7$

Solve for $k$:

$k = \frac{58.8}{7} = 8.4$

So, the constant of proportionality is:

$k = 8.4$

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Step 2: Write the variation equation in terms of $x$

Now that we know $k = 8.4$, the variation equation can be written as:

$y = 8.4x$

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Step 3: Find $y$ when $x = 32$

Using the equation $y = 8.4x$, substitute $x = 32$ to find $y$:

$y = 8.4 \times 32 = 268.8$

So, when $x = 32$, $y = 268.8$.

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Final Answers:

1. Constant of proportionality $k$: $8.4$
2. Variation equation: $y = 8.4x$
3. Value of $y$ when $x = 32$: $268.8$

Would you like further details on these steps, or do you have any questions?

Here are 5 related questions to explore:

1. How do you solve inverse variations?
2. What are some real-life examples of direct variations?
3. How can the variation equation be applied in geometry?
4. Can the constant of proportionality be negative? How would the equation change?
5. How do you find $x$ given a value for $y$ in a variation equation?

Tip: In direct variation, always remember that the constant $k$ remains the same for all values of $x$ and $y$ in the equation.