The problem asks to calculate the volume of a rectangular prism, given the areas of its three adjacent faces.

The areas provided are:

• 6 m²
• 8 m²
• 12 m²

Let the dimensions of the prism be $x$, $y$, and $z$. These represent the length, width, and height of the rectangular prism, respectively.

From the given face areas, we know:

• $xy = 6$ m² (area of the face formed by dimensions $x$ and $y$)
• $xz = 8$ m² (area of the face formed by dimensions $x$ and $z$)
• $yz = 12$ m² (area of the face formed by dimensions $y$ and $z$)

To find the volume $V$ of the prism, we use the formula: $V = x \cdot y \cdot z$

We can find the value of $x$, $y$, and $z$ by solving these equations. First, multiply all three equations together: $(xy)(xz)(yz) = (6)(8)(12)$ $x^2 y^2 z^2 = 576$ Taking the square root of both sides: $xyz = \sqrt{576} = 24$

Thus, the volume of the rectangular prism is: $V = 24 \, \text{m}^3$

Would you like further details on the calculations or clarification on any steps?

Here are some related questions you might want to explore:

1. How do we solve for individual dimensions $x$, $y$, and $z$?
2. Can we use the Pythagorean theorem in a rectangular prism's diagonal?
3. How would the volume change if one face area were doubled?
4. What is the surface area of the rectangular prism?
5. How would the volume change if one of the dimensions were halved?

Tip: When solving for volume using face areas, multiplying all equations and taking square roots can save time!