Calculate Cuboid Volume: Given Surface Area 450x cm² and Side Length 7

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.