Calculate the Volume-to-Area Ratio: Cuboid (3×2×4) vs Triangle ABK

To solve the problem, let's proceed with these steps:

• Step 1: Calculate the volume of the cuboid.

• Step 2: Determine the area of triangle ABK.

• Step 3: Compute the ratio between the two quantities.

Now, let's execute these steps:
Step 1: The volume of the cuboid is given by the formula:

$V = l \times w \times h$

Substituting the given dimensions:

$V = 3 \times 2 \times 4 = 24 \text{ cm}^3$

Step 2: To find the area of triangle ABK, we first need to determine its base and height. Given the dimensions, we consider segment AB as the height and segment BK as the base. Thus, base BK is the width, 2 cm, and height AB is the full height, 4 cm. The area is calculated by:

$\text{Area of } \triangle ABK = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 4 = 4 \text{ cm}^2$

Step 3: With these calculations, the ratio between the volume of the cuboid and the area of triangle ABK is:

$\text{Ratio} = \frac{\text{Volume of Cuboid}}{\text{Area of } \triangle ABK} = \frac{24}{4} = 6$

Therefore, the ratio between the volume of the orthocahedron and the area of triangle ABK is $6$.