Surface Area of a Cuboid: Finding Area based off Perimeter and Vice Versa

The first step is to calculate the relevant data for all the components of the box.

The length of the box = 6

Given that the height of a cuboid is equal to half its length we are able to deduce the height of the box as follows : 6/2= 3

Hence the height = 3

In order to determine the width, we insert the known data into the formula for the volume of the box:

height*length*width = volume of the cuboid.

3*6*width = 72

18*width=72

We divide by 18:

Hence the width = 4

We are now able to return to the initial question regarding the surface of the cuboid.

Remember that the formula for the surface area is:

(height*length+height*width+length*width)*2

We insert the known data leaving us with the following result:

(3*6+4*3+4*6)*2=

(12+24+18)*2=

(54)*2=

108

To solve this problem, we'll use the formulas related to a cuboid:

• Step 1: Understand that the surface area is given by $2(ab + bc + ca) = 450x$, where one side is $a = 7$.
• Step 2: Solve for $b$ and $c$ in terms of $a$ and total surface area.
• Step 3: Calculate the volume $V = a \times b \times c$.

Let's proceed with the solution:

Given that the surface area $2(ab + bc + ca) = 450x$ and $a = 7$, we substitute $a$ in the equation:
$2(7b + 7c + bc) = 450x$.
Simplify: $14b + 14c + 2bc = 450x$.
$7b + 7c + bc = 225x$. (after dividing by 2)

Next, express volume $V = a \times b \times c = 7 \times b \times c$.
We know from the surface area problem: $b + c + \frac{bc}{7} = 225x/7$.

Plug in $b+c = u$ and rearrange in terms of quadratic:
$(bc) = 7(\frac{225x}{7} - u)$. Thus, $bc = 225x - 7u$.
The assumed equation is $b+c = 14$ and $bc = 225x - 49$ by substituting obtained relations.

Thus the volume finally is:

$V = 7 \times (225x - 49) = 7 \times bc$.
Hence, results in calculating $bc = 225x - 49$.

Therefore, the volume of the cuboid is $225x - 49$ cm³.

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 $201.6 \, \text{cm}^3$.