Calculate Surface Area: Cuboid with 7cm Side and Percentage-Based Dimensions

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 $\textbf{272.538 cm}^2$.