Volume of a Orthohedron: A shape consisting of several shapes (requiring the same formula)

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 $96 \, \text{cm}^3$.

Let's solve the problem by calculating the volume of the large cuboid step-by-step:

Step 1: Determine the dimensions of each small orthohedron

• Given $AB = 5$ and $BC = 4$, we understand these are the sides of a triangle segment within the cuboid's formation.
• The problem states $DB = \frac{1}{3}$ of the total sum of $AB$ and $CB$, which implies $CB$ is as large as $BC$.
• Hence, $DB = \frac{1}{3} (5 + 4) = \frac{1}{3} \times 9 = 3$.
• This deduction allows us to assume the height of each small orthohedron $h = 3$.

Step 2: Calculating the volume of one small orthohedron

• Each orthohedron has dimensions: $AB = 5$, $BC = 4$, and the height $h = 3$.
• Therefore, the volume $V_{\text{small}}$ is calculated as:
• $V_{\text{small}} = AB \times BC \times h = 5 \times 4 \times 3 = 60$ cm³

Step 3: Calculate the total volume of the large cuboid

• The large cuboid is composed of 5 such small orthohedra, so:
• $V_{\text{large}} = 5 \times V_{\text{small}} = 5 \times 60 = 300$ cm³

Thus, the volume of the large cuboid is $300$ cm³.

To solve this problem, we'll follow these steps:

• Step 1: Identify the volume formula and apply the given dimensions
• Step 2: Calculate the total volume by considering the number of orthohedra

Let's proceed:

Step 1: For a single orthohedron with dimensions of 5 meters, 6 meters, and 8 meters, the volume is calculated using:

$\text{Volume of one orthohedron} = 5 \times 6 \times 8$

This equates to $5 \times 6 = 30$, and then $30 \times 8 = 240$. Therefore, the volume of one orthohedron is $240 \, \text{cubic meters}$.

Step 2: Since there are five such orthohedra forming the complete body, multiply the volume of one by 5:

$\text{Total volume} = 5 \times 240 = 1200 \, \text{cubic meters}$

Thus, the total volume of the created body is $1200 \, \text{cubic meters}$.

The correct choice that corresponds to this volume is:

• $1200 \, \text{cubic meters}$

Therefore, the solution to the problem is $1200$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the volume of a single cuboid using the dimensions provided.
• Step 2: Multiply the volume of one cuboid by the number of cuboids to find the total volume of the assembled body.

Let's work through each step:

Step 1: The dimensions of the given cuboid are 3 meters, 4 meters, and 7 meters. The volume of this cuboid is calculated using the formula:

$\text{Volume of one cuboid} = \text{length} \times \text{width} \times \text{height} = 3 \times 4 \times 7$

Performing the calculation:

$3 \times 4 = 12$

$12 \times 7 = 84$

Therefore, the volume of one cuboid is $84 \text{ cubic meters}$.

Step 2: Since there are three identical cuboids combined to form the body, we multiply the volume of one cuboid by 3:

$\text{Total Volume} = 84 \times 3$

Carrying out the multiplication:

$84 \times 3 = 252$

Therefore, the volume of the created body is $\text{252 cubic meters}$.

Thus, the correct answer is $252$, which matches choice 3 in the given options.

To solve this problem, we will calculate the volume of each rectangular prism separately and then sum these volumes:

• Rectangular Prism 1: The prism with a square base has dimensions as follows:
• Side length of the square base = 5 units
• Height = 9 units
• The volume of a rectangular prism with a square base is given by $V = \text{side}^2 \times \text{height}$.
• Substituting the given values: $V_1 = 5^2 \times 9 = 25 \times 9 = 225 \text{ cubic units}$.
• Rectangular Prism 2: The second prism's dimensions:
• Length = 3 units
• Width = 2 units
• Height = 4 units
• The volume of this rectangular prism is calculated as $V = \text{length} \times \text{width} \times \text{height}$.
• Substituting the given values: $V_2 = 3 \times 2 \times 4 = 24 \text{ cubic units}$.

Finally, adding the volumes of the two prisms gives us the total volume:

$V_{\text{total}} = V_1 + V_2 = 225 + 24 = 249 \text{ cubic units}$.

Therefore, the volume of the new shape is $249$ cubic units.

To solve this problem of calculating the volume of a new shape formed by three identical cuboids, we need to follow these steps:

• Step 1: Calculate the volume of a single cuboid.
• Step 2: Multiply the volume of one cuboid by the number of cuboids (three) to get the total volume.

Let's work through these steps:

Step 1: The volume of a single cuboid is calculated using the formula for the volume of a cuboid, which is:
$V = \text{length} \times \text{width} \times \text{height}$

Given that the length and width are $5$ meters and $7$ meters, and assuming the height (expected to be the same as the base's side since it's square-based and not otherwise specified) is another $5$, we have:
$V_{\text{cuboid}} = 5 \times 7 \times 5 = 175 \, \text{m}^3$

Step 2: Multiply the volume of one cuboid by three because there are three identical cuboids:
$V_{\text{total}} = 3 \times 175 = 525 \, \text{m}^3$

Therefore, the volume of the new shape is $\mathbf{525} \, \text{m}^3$.

To solve this problem, we'll follow these steps:

• Step 1: Determine individual cuboid dimensions and calculate its volume
• Step 2: Calculate the cumulative volume resulting from the merger of two identical cuboids

Let's work through these steps:
Step 1: The given dimensions for a square-based cuboid suggest each has a base of 4 meters and a height of 8 meters. Therefore, the volume of one cuboid is determined using the formula: $V = 4 \times 4 \times 8$.

Calculating this gives: $V = 128 \text{ cubic meters}$.
Step 2: Since two such cuboids are combined to form the bodand, the total volume is: $2 \times 128 = 256 \text{ cubic meters}$.

Therefore, the volume of the created bodand is $256 \text{ cubic meters}$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the volume of one smaller cuboid.
• Step 2: Multiply the volume of one smaller cuboid by the number of cuboids (5) to find the total volume of the large cuboid.

Now, let's work through each step:
Step 1: The dimensions for each smaller cuboid are given as:

• Length: $4 \, \text{cm}$
• Width: $2 \, \text{cm}$
• Height: $5 \, \text{cm}$

Using the volume formula for a cuboid, we find the volume of one smaller cuboid:
$V_{\text{small}} = 4 \times 2 \times 5 = 40 \, \text{cm}^3$
Step 2: Since there are 5 such smaller cuboids, the total volume for the large cuboid is:
$V_{\text{large}} = 5 \times 40 = 200 \, \text{cm}^3$

Therefore, the volume of the large cuboid is $200 \, \text{cm}^3$.

To solve this problem, we need to determine the volume of a small cuboid when given a total volume comprising four such cuboids.

First, let's establish the dimensions of the large cuboid. According to the problem:

• The length ($L$) of the large cuboid is $10 \, \text{cm}$.
• The width ($W$) is half the length, so $W = \frac{10}{2} = 5 \, \text{cm}$.
• The height ($H$) is $\frac{4}{5}$ of the length, so $H = \frac{4}{5} \times 10 = 8 \, \text{cm}$.

Next, calculate the volume of the large cuboid using the volume formula:

$\text{Volume of Large Cuboid} = L \times W \times H = 10 \, \text{cm} \times 5 \, \text{cm} \times 8 \, \text{cm} = 400 \, \text{cm}^3$

Since the large cuboid is composed of 4 identical smaller cuboids, the volume of each smaller cuboid is:

$\text{Volume of Small Cuboid} = \frac{\text{Volume of Large Cuboid}}{4} = \frac{400 \, \text{cm}^3}{4} = 100 \, \text{cm}^3$

Thus, the volume of one small cuboid is $100 \, \text{cm}^3$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the volume of one prism of each type using the formula for volume: $\text{Volume} = \text{length} \times \text{width} \times \text{height}$.
• Step 2: Multiply each result by two since two identical prisms form each body.
• Step 3: Sum the volumes of the two bodies to get the total volume of the combined structure.

Now, let's work through each step:

Step 1: Calculate the volume of Prism 1:
For Prism 1 with dimensions $6 \, \text{m} \times 7 \, \text{m} \times 3 \, \text{m}$, the volume is:

$\text{Volume of one Prism 1} = 6 \times 7 \times 3 = 126 \, \text{m}^3$.

Step 2: Since the body is made from two Prism 1 structures, multiply by 2:

$\text{Total Volume of first body} = 2 \times 126 = 252 \, \text{m}^3$.

Step 3: Calculate the volume of Prism 2:
For Prism 2 with dimensions $3 \, \text{m} \times 5 \, \text{m} \times 4 \, \text{m}$, the volume is:

$\text{Volume of one Prism 2} = 3 \times 5 \times 4 = 60 \, \text{m}^3$.

Since the second body is made from two Prism 2 structures, multiply by 2:

$\text{Total Volume of second body} = 2 \times 60 = 120 \, \text{m}^3$.

Finally, add the two total volumes together to find the volume of the combined structure:

$\text{Total Volume} = 252 + 120 = 372 \, \text{m}^3$.

Therefore, the volume of the resulting body is $372 \, \text{m}^3$.