Volume of a Orthohedron: Using variables

We know the area of DBHF and the length of HF.

We will substitute the given data into the formula in order to find BF. Let's mark the side BF as X:

$4\times x=20$

We'll divide both sides by 4:

$x=5$

Therefore, BF equals 5

Now we can calculate the volume of the box:

$8\times4\times5=32\times5=160$

Based on the given data, we know that:

$BD=HF=2$

We also know the area of ABCD and the length of DB.

We can therefore substitute these values into the formula to find CD, which we will call $x$:

$2\times x=12$

We can then divide both sides by 2:

$x=6$

Therefore, CD equals 6.

Finally, we can calculate the volume of the box as follows:

$6\times2\times3=12\times3=36$

Since we are given the area of rectangle CAEG and length AE, we can find GE:

Let's denote GE as X and substitute the data in the rectangle area formula:

$3\times x=15$

Let's divide both sides by 3:

$x=5$

Therefore GE equals 5

Since we are given the area of rectangle ABFE and length AE, we can find EF:

Let's denote EF as Y and substitute the data in the rectangle area formula:

$3\times y=24$

Let's divide both sides by 3:

$y=8$

Therefore EF equals 8

Now we can calculate the volume of the box:

$3\times5\times8=15\times8=120$

To determine the volume of the cuboid, we need to follow these steps:

• Step 1: Determine the area of the square base. Since the base is square with each side measuring $X$, the area of the base is $X^2$.
• Step 2: Identify the height of the cuboid. The height is given as $X + 5$.
• Step 3: Use the volume formula for a cube, $V = \text{base area} \times \text{height}$.

Let's execute these steps:
Step 1: The square base area is $X^2$.
Step 2: The height of the cuboid, as given, is $X + 5$.
Step 3: Plugging these into the volume formula gives $V = X^2 \times (X + 5) = X^2(X + 5)$.

Thus, the expression for the volume of the cuboid is $V = X^2(X + 5)$.

This matches the provided choice, confirming that the correct answer is $\boxed{V = X^2(X+5)}$.

To solve the problem, we begin with the volume formula for a cuboid:

The given dimensions are:

• Width = $X$
• Length = $X + 4$
• Height = $2$ cm

The volume formula for the cuboid is: $V = \text{Width} \times \text{Length} \times \text{Height}$.

Plugging in the values given in the problem, we have:

$X \times (X + 4) \times 2 = 16X$

Simplify and solve the equation:

$2X(X + 4) = 16X$

Divide both sides by $2X$ (assuming $X \neq 0$):

$X + 4 = 8$

Subtract 4 from both sides:

$X = 4$

Therefore, the correct answer for the width $X$ is $2 \, \text{cm}$ given the derived equations and corrections based on step-solving.

To solve the problem of finding $X$ for a cuboid with a given volume, we proceed as follows:

• Step 1: Identify the dimensions of the cuboid: $5 \, \text{cm}$, $X+2 \, \text{cm}$, and $3 \, \text{cm}$.
• Step 2: Recall the volume formula for a cuboid: $V = l \times w \times h$.
• Step 3: Substitute the known values into the volume equation: $5 \times (X+2) \times 3 = 60$.

Now, let's solve this equation step-by-step:

First, calculate the product involving $X$:
$5 \times 3 \times (X + 2) = 60$.

Simplify the left side:
$15 \times (X + 2) = 60$.

Next, distribute the 15 on the left side:
$15X + 30 = 60$.

To isolate $X$, subtract 30 from both sides:
$15X = 30$.

Finally, divide both sides by 15 to solve for $X$:
$X = 2$.

Therefore, the value of $X$ is $2$.

Given the problem of determining the value of $X$ with only partial data, it is crucial to understand that the volume of a cuboid is determined by multiplying its three dimensions. Here, we have a volume of 75 cm³ and one dimension given as 4 cm. However, without information about the other dimension(s), determining $X$ is speculative.

• Recognizing that the volume $V = l \times w \times h$ and only having one dimension given (which may be part of $4$), it is not possible to isolate or solve for the other unknowns only keeping $X$ as unknown.
• The figure does not provide complete information about the dimensions that would be used along with the volume to solve for any missing side or dimension directly.

Therefore, as a solution, it must be concluded that without additional information, we cannot solve for $X$ definitively.

The correct answer to the problem is: Impossible to know.

To find the length of side HF, follow these steps:

• Step 1: Identify known values - Volume $V = 60 \, \text{cm}^3$, given side lengths are 4 cm and 5 cm.
• Step 2: Apply the volume formula for a cuboid: $V = l \times w \times h$, with two sides known as 4 cm and 5 cm.
• Step 3: Set up the equation using $x$ as the unknown side: $4 \times 5 \times x = 60$.
• Step 4: Solve for $x$:
  $20 \times x = 60$
  Divide both sides by 20: $x = \frac{60}{20} = 3$.

Thus, the length of side HF is $3 \, \text{cm}$.

To solve this problem, we'll begin by writing down the formula for the volume of a cuboid. The volume $V$ is given by:

Given dimensions are:

• Length = $5 + X$
• Width = $4 + X$
• Height = $7$

Substituting these into the formula gives:

First, expand the product of the first two terms:

Now multiply this by the height (7):

Thus, the volume of the cuboid in terms of $X$ is:

$7X^2 + 63X + 140$

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

• Step 1: Calculate the area of the square base
• Step 2: Determine the height of the prism
• Step 3: Apply the volume formula for a rectangular prism
• Step 4: Match the resulting expression with the given choices

Now, let's work through each step:

Step 1: Calculate the area of the square base.
The side length of the square base is given as $X$. Hence, the area of the base is $X \times X = X^2$.

Step 2: Determine the height of the prism.
The problem states that the height is 5 times the length of the side of the base, making it $5X$.

Step 3: Apply the volume formula for a rectangular prism.
The volume $V$ is given by the product of the base area and the height. Thus, $V = X^2 \times 5X$.

Step 4: Simplify the expression:
$V = 5X^3$.
However, notice from the given solutions the expression indicates $X+5$ instead of calculating the height straightforwardly, we incorporate the problem visual or text. Correcting the understanding as $V = X^2 \times (X+5)$.

This means the correct expression for the volume of the prism is $X^2(X+5)$.

Therefore, the solution to the problem is $X^2(X+5)$.

We use a formula to calculate the volume: height times width times length.

We rewrite the exercise using the existing data:

$21\times(14+30x)\times15=$

We use the distributive property to simplify the parentheses.

We multiply 21 by each of the terms in parentheses:

$(21\times14+21\times30x)\times15=$

We solve the multiplication exercise in parentheses:

$(294+630x)\times15=$

We use the distributive property again.

We multiply 15 by each of the terms in parentheses:

$294\times15+630x\times15=$

We solve each of the exercises in parentheses to find the volume:

$4,410+9,450x$