Understanding the Cuboid Problem

This problem asks us to find the volume of a cuboid given the areas of three faces that meet at a single vertex. A cuboid is a three-dimensional shape with six rectangular faces. When three faces meet at a vertex, they are mutually perpendicular. Let the dimensions of the cuboid be length ($\text{l}$), width ($\text{w}$), and height ($\text{h}$).

The areas of the three faces sharing a vertex will be the products of two dimensions: $l \times w$, $w \times h$, and $h \times l$.

Setting up Equations for Cuboid Face Areas

We are given that the areas of these three faces are 20 $\text{m}^2$, 32 $\text{m}^2$, and 40 $\text{m}^2$. We can set up the following equations:

1. Area 1: $\text{lw} = 20 \text{ m}^2$
2. Area 2: $\text{wh} = 32 \text{ m}^2$
3. Area 3: $\text{hl} = 40 \text{ m}^2$

Our goal is to find the volume of the cuboid, which is given by the formula $\text{V} = \text{lwh}$.

Calculating the Volume of the Cuboid

To find the volume $\text{lwh}$, we can multiply the three equations we set up:

$(\text{lw}) \times (\text{wh}) \times (\text{hl}) = 20 \times 32 \times 40$

Multiplying the terms on the left side gives us:

$\text{l}^2\text{w}^2\text{h}^2$

Multiplying the numbers on the right side gives us:

$20 \times 32 \times 40 = 640 \times 40 = 25600$

So, we have:

$\text{l}^2\text{w}^2\text{h}^2 = 25600$

Notice that $\text{l}^2\text{w}^2\text{h}^2$ is the same as $(\text{lwh})^2$. So, we have:

$(\text{lwh})^2 = 25600$

To find the volume $\text{lwh}$, we need to take the square root of both sides of the equation:

$\text{lwh} = \sqrt{25600}$

Calculating the square root:

$\sqrt{25600} = \sqrt{256 \times 100} = \sqrt{256} \times \sqrt{100} = 16 \times 10 = 160$

So, the volume of the cuboid is 160 $\text{m}^3$.

The volume is $\text{V} = 160 \text{ m}^3$.

Verification of the Cuboid Volume

Let's quickly check if this volume is consistent with the given face areas. If $lwh = 160$, we can potentially find $l, w, h$:

• $\text{lw} = 20 \Rightarrow \text{h} = \frac{\text{lwh}}{\text{lw}} = \frac{160}{20} = 8$
• $\text{wh} = 32 \Rightarrow \text{l} = \frac{\text{lwh}}{\text{wh}} = \frac{160}{32} = 5$
• $\text{hl} = 40 \Rightarrow \text{w} = \frac{\text{lwh}}{\text{hl}} = \frac{160}{40} = 4$

With $l=5$, $w=4$, and $h=8$, let's check the face areas:

• $\text{lw} = 5 \times 4 = 20$ (Matches)
• $\text{wh} = 4 \times 8 = 32$ (Matches)
• $\text{hl} = 8 \times 5 = 40$ (Matches)

The dimensions $l=5 \text{ m}$, $w=4 \text{ m}$, $h=8 \text{ m}$ give the correct face areas, and their product is $5 \times 4 \times 8 = 160$, which is the volume.

Final Answer

The volume of the cuboid is 160 $\text{m}^3$. This matches option 3.

Revision Table: Cuboid Calculations

Additional Information: Cuboid Properties

A cuboid is a fundamental 3D geometric shape. Here are some key properties:

• It has 6 faces, all of which are rectangles.
• Opposite faces are identical (congruent).
• It has 12 edges.
• It has 8 vertices.
• The faces meeting at a vertex are perpendicular to each other.
• A special type of cuboid where all edges are equal is a cube.
• The surface area of a cuboid is the sum of the areas of all six faces: $2(\text{lw} + \text{wh} + \text{hl})$.

Knowing the areas of three mutually perpendicular faces is a direct way to find the volume by using the relationship $(\text{lwh})^2 = (\text{lw})(\text{wh})(\text{hl})$.