The given equation is:

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

Where:

• $V$ is the volume,
• $a$ is the edge length of the hexagonal base,
• $h$ is the height of the prism.

We want to solve for $a$ by rearranging this equation.

Step 1: Isolate the square root term

Start by isolating the square root term on one side of the equation: $V = 33 - \sqrt{2a^2h}$

Subtract 33 from both sides: $V - 33 = -\sqrt{2a^2h}$

Multiply both sides by -1: $33 - V = \sqrt{2a^2h}$

Step 2: Square both sides

To eliminate the square root, square both sides: $(33 - V)^2 = 2a^2h$

Step 3: Solve for $a^2$

Now, solve for $a^2$ by dividing both sides by $2h$: $a^2 = \frac{(33 - V)^2}{2h}$

Step 4: Solve for $a$

Finally, take the square root of both sides to find $a$: $a = \sqrt{\frac{(33 - V)^2}{2h}}$

Final Result

This is the equation for $a$ in terms of $V$ and $h$:

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

Or equivalently:

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

This is the equation that you need for $a$. Looking at the given options, the one that matches this form is:

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