Calculate Cuboid Length: Surface Area 406 cm² with 7 cm Square Face

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

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: The problem gives us a square face of the cuboid with side length 7 cm, meaning two dimensions are $7 \text{ cm}$ each. The total surface area of the cuboid is 406 cm².

Step 2: We'll use the formula for the surface area of a cuboid:
$\text{Surface area} = 2(lw + lh + wh)$

Let the dimensions be $l$ (length), $w = 7 \, \text{cm}$ (width), and $h = 7 \, \text{cm}$ (height).

The formula becomes:
$2(l \times 7 + l \times 7 + 7 \times 7) = 406$

Step 3: Simplify and solve for $l$.
Plug in the known values:
$2(7l + 7l + 49) = 406$

Simplify:
$2(14l + 49) = 406$

Divide by 2:
$14l + 49 = 203$

Subtract 49 from both sides:
$14l = 154$

Divide by 14:
$l = \frac{154}{14}$

So,
$l = 11 \, \text{cm}$

Therefore, the length of the cuboid is 11 cm.