Volume of a Orthohedron: Finding Area based off Perimeter and Vice Versa

To solve this problem, we are going to determine the volume of a cuboid given its length, width, and overall surface area. Here's how we will do it:

• Firstly, calculate the height $h$ of the cuboid using the surface area formula.
• Then, use the calculated height to determine the volume of the cuboid.

Now, let's work through each step:

Step 1: Calculate the height $h$.

We know the surface area of a cuboid is given by the formula:

$2(lw + lh + wh) = \text{Surface Area}$

Substituting the known values:

$2(5 \cdot 4 + 5 \cdot h + 4 \cdot h) = 94$

Simplify:

$2(20 + 5h + 4h) = 94$ $2(20 + 9h) = 94$

Divide both sides by 2:

$20 + 9h = 47$

Simplify further to solve for $h$:

$9h = 27$ $h = 3 \, \text{cm}$

Step 2: Calculate the volume using the height $h$.

Now that we know $h = 3 \, \text{cm}$, use the volume formula for the cuboid:

$\text{Volume} = l \times w \times h$ $\text{Volume} = 5 \times 4 \times 3 = 60 \, \text{cm}^3$

Therefore, the volume of the cuboid is $\mathbf{60 \, \text{cm}^3}$.

To solve this problem, follow these steps:

• Step 1: Identify the given information: $l = 8 \, \text{cm}$, $w = 4 \, \text{cm}$, and $S = 136 \, \text{cm}^2$.
• Step 2: Use the surface area formula to find height $h$: $136 = 2(8 \cdot 4 + 8 \cdot h + 4 \cdot h)$.
• Step 3: Simplify and solve for $h$.
• Step 4: Use the volume formula $V = l \cdot w \cdot h$.
• Step 5: Substitute the value of $h$ into the volume formula.
• Step 6: Calculate and obtain the final result.

Now, let's calculate:

Starting with the surface area equation:

$136 = 2(8 \cdot 4 + 8 \cdot h + 4 \cdot h)$.

Simplifying gives:

$136 = 2(32 + 8h + 4h)$.

$136 = 2(32 + 12h)$.

$136 = 64 + 24h$.

Subtract 64 from both sides:

$72 = 24h$.

Divide both sides by 24:

$h = 3 \, \text{cm}$.

Now, calculate the volume using $V = l \cdot w \cdot h$:

$V = 8 \cdot 4 \cdot 3$.

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

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

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

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula to find the unknown dimension
• Step 3: Calculate the volume using the known dimensions

Now, let's work through each step:
Step 1: We know the surface area $S = 240 \, \text{cm}^2$, and two dimensions: 12 cm and 3 cm.

Step 2: The formula for the surface area of a rectangular prism is:
$S = 2(lw + lh + wh)$
Substituting the known values into the equation:
$240 = 2(12 \times 3 + 12 \times h + 3 \times h)$

Simplify and solve for $h$:
$240 = 2(36 + 12h + 3h)$
$240 = 2(36 + 15h)$
$240 = 72 + 30h$
$168 = 30h$
$h = \frac{168}{30} = 5.6 \, \text{cm}$

Step 3: Now that we know all dimensions, use the volume formula:
$V = l \times w \times h = 12 \times 3 \times 5.6$

Perform the calculation:
$V = 36 \times 5.6 = 201.6 \, \text{cm}^3$

Therefore, the volume of the rectangular prism is $201.6 \, \text{cm}^3$.