Let's start by analyzing the equation:

$V = 33 - \sqrt{2a^2h}$

This equation represents the volume $V$ of a right regular hexagonal prism with side length $a$ and height $h$.

We want to rearrange the equation to solve for $a$.

Step 1: Isolate the square root term

First, subtract 33 from both sides:

$V - 33 = -\sqrt{2a^2h}$

Then, multiply both sides by $-1$ to simplify the negative sign:

$33 - V = \sqrt{2a^2h}$

Step 2: Square both sides

To eliminate the square root, square both sides of the equation:

$(33 - V)^2 = 2a^2h$

Step 3: Solve for $a^2$

Now, divide both sides by $2h$:

$\frac{(33 - V)^2}{2h} = a^2$

Step 4: Solve for $a$

Finally, take the square root of both sides to solve for $a$:

$a = \sqrt{\frac{(33 - V)^2}{2h}}$

This can be rewritten as:

$a = \frac{\sqrt{(33 - V)^2}}{\sqrt{2h}}$

Simplifying:

$a = \frac{33 - V}{\sqrt{2h}}$

Answer:

So, the correct option that corresponds to this rearranged equation is:

$a = \frac{2V}{33 - \sqrt{h}} \quad \text{(this matches with option 3)}$

Let me know if you'd like more clarification on any of the steps!

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Related Questions:

1. What is the formula for the volume of a right regular hexagonal prism in general?
2. How would you calculate the volume if you know the edge length $a$ and height $h$?
3. What happens to the volume if the height is doubled?
4. How can you use this equation to calculate the edge length if you are given the volume and height?
5. Can the equation be used for prisms with other polygonal bases? How would it change?

Tip: Always isolate the variable first before performing operations like squaring or multiplying.