Volume of a Orthohedron: Using ratios for calculation

To solve the problem, let's determine the dimensions of the large cuboid:

• Step 1: Identify the given values: $AB = 4$, $BC = \frac{3}{4} \times AB$, and $BE = \frac{1}{2} \times AB$.
• Step 2: Calculate the dimensions of the cuboid:
• Since $AB = 4$, we find $BC = \frac{3}{4} \times 4 = 3$.
• Next, $BE = \frac{1}{2} \times 4 = 2$.
• Step 3: Calculate the volume of the large cuboid using the formula:
• $\text{Volume} = AB \times BC \times BE = 4 \times 3 \times 2 = 24$.
• Since the large cuboid is composed of four equal orthohedra, the volume of the entire large cuboid is $4 \times 24 = 96 \, \text{cm}^3$.

So, the volume of the large cuboid is $96 \, \text{cm}^3$.

To solve this problem, we need to find the length of the cuboid using its volume and given dimensions. Let's break this down:

• Step 1: Express the length and width using the ratio given. Let $l = 3x$ and $w = 2x$, where $x$ is a common factor.
• Step 2: Apply the formula for the volume of the cuboid, $V = l \times w \times h$. Given that $h = 4$ cm and $V = 96$ cm³, the equation becomes:

Now, simplify and solve for $x$ :

Dividing both sides by 24 gives:

Taking the square root of both sides, we find:

Thus, the length of the cuboid is:

Therefore, the solution to the problem is $6 \text{ cm}$.

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 $6$.