Surface Area of a Cuboid: Calculation using percentages

First, we'll express $b$ and $c$ in terms of $a$:
$b = 1.2a \quad \text{and} \quad c = 1.44a$

Next, substitute these into the surface area formula for a cuboid:

$2(ab + bc + ca) = 436$

$ab = a \cdot (1.2a) = 1.2a^2$

$bc = (1.2a) \cdot (1.44a) = 1.728a^2$

$ca = (1.44a) \cdot a = 1.44a^2$

Substitute these into the formula:

$2(1.2a^2 + 1.728a^2 + 1.44a^2) = 436$

Simplify inside the parentheses:

$2 \cdot 4.368a^2 = 436$

$8.736a^2 = 436$

Solve for $a^2$:

$a^2 = \frac{436}{8.736}$

$a^2 \approx 49.93$

Taking the square root, we find:

$a \approx \sqrt{49.93} \approx 7.06$

Now, find $b$ and $c$:

$b = 1.2a = 1.2 \times 7.06 \approx 8.47$

$c = 1.44a = 1.44 \times 7.06 \approx 10.17$

Therefore, the dimensions of the orthohedron are:

$a = 7.06$, $b = 8.47$, and $c = 10.17$.

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

• Step 1: Identify the dimensions of the cuboid using the modifications given.
• Step 2: Apply the surface area formula for the cuboid using these dimensions.

Now, let's work through each step:
Step 1: We are given that the first side length is $7$ cm. The second side is $10\%$ smaller than the first side. Calculate the second side length:
$\text{Second Side} = 7\,\text{cm} - 0.1 \times 7\,\text{cm} = 7\,\text{cm} - 0.7\,\text{cm} = 6.3\,\text{cm}$

The third side is $10\%$ larger than the second side. Calculate the third side length:
$\text{Third Side} = 6.3\,\text{cm} + 0.1 \times 6.3\,\text{cm} = 6.3\,\text{cm} + 0.63\,\text{cm} = 6.93\,\text{cm}$

Step 2: Now calculate the surface area using the formula $2(lw + lh + wh)$:
Let $l = 7\,\text{cm}$, $w = 6.3\,\text{cm}$, and $h = 6.93\,\text{cm}$. Substitute into the formula:
$\text{Surface Area} = 2 \left( 7 \times 6.3 + 7 \times 6.93 + 6.3 \times 6.93 \right)$

Calculate each part:
$7 \times 6.3 = 44.1$
$7 \times 6.93 = 48.51$
$6.3 \times 6.93 = 43.659$

Add these results together:
$44.1 + 48.51 + 43.659 = 136.269$

Thus, the surface area is:
$2 \times 136.269 = 272.538 \,\text{cm}^2$

Therefore, the solution to the problem is $\textbf{272.538 cm}^2$.