Question
Question \( 1-2 \) Look st the equation \[ V=\frac{3 \sqrt{3}}{2} a^{2} h \] The equation represents the volume of a night regular hexagonal prism where each edge of the hexa a, by rearranging the equation shown? \( a=\frac{\sqrt{2 V}}{3 \sqrt{3 h}} \) \( a=\frac{\sqrt{2 V h}}{3 \sqrt{3 h}} \) \( a=\sqrt{\frac{2 V}{3 \sqrt{3 h}}} \)
Ask by Park Boone. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( a = \sqrt{\frac{2V}{3 \sqrt{3} h}} \)
Solution
To solve for \( a \) in the equation for the volume of a right regular hexagonal prism:
\[
V = \frac{3 \sqrt{3}}{2} a^{2} h
\]
**Step-by-Step Solution:**
1. **Start with the original equation:**
\[
V = \frac{3 \sqrt{3}}{2} a^{2} h
\]
2. **Isolate \( a^{2} \):**
Multiply both sides by \( \frac{2}{3 \sqrt{3}} \):
\[
a^{2} = \frac{2V}{3 \sqrt{3} h}
\]
3. **Take the square root of both sides to solve for \( a \):**
\[
a = \sqrt{\frac{2V}{3 \sqrt{3} h}}
\]
**Final Answer:**
\[
a = \sqrt{\frac{2V}{3 \sqrt{3} h}}
\]
This corresponds to the **third option** provided:
\[
a = \sqrt{\frac{2 V}{3 \sqrt{3} h}}
\]
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Beyond the Answer
The equation provided represents the volume of a regular hexagonal prism, and to solve for \( a \), we need to isolate \( a \) from the equation. We can begin by multiplying both sides of the equation by \( \frac{2}{3 \sqrt{3} h} \) and then simplifying terms to express \( a \) in terms of \( V \) and \( h \). You'll want to keep track of your algebraic manipulation—the rearrangement technique is your best friend here! When performed correctly, to isolate \( a \), we arrive at the equation \( a = \sqrt{\frac{2V}{3\sqrt{3}h}} \). The key here is careful rearrangement: always ensure your steps maintain the integrity of the equation’s operations!
