Calculate Cuboid Surface Area: Using 1:2:4 Ratio with 5cm Middle Length

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

• Step 1: Solve for the variable $x$ using the given ratio and dimension.
• Step 2: Calculate the dimensions of the cuboid.
• Step 3: Apply the surface area formula for a cuboid.
• Step 4: Carry out the calculations to find the total surface area.

Now, let's work through each step:

Step 1: Solving for $x$

We know that the dimensions of the cuboid are in the ratio $1:2:4$. Assigning the dimensions as $x$, $2x$, and $4x$, and knowing that the middle dimension is 5 cm, we have $2x = 5$. Solving for $x$, we get:

$x = \frac{5}{2} = 2.5 \text{ cm}$

Step 2: Calculate the dimensions

Now, using the value of $x$:

• First dimension: $x = 2.5 \text{ cm}$
• Second dimension: $2x = 5 \text{ cm}$
• Third dimension: $4x = 10 \text{ cm}$

Step 3: Apply the surface area formula

The formula for the surface area of a cuboid is:

$A = 2(lw + lh + wh)$

Substitute the dimensions into the formula:

$A = 2(2.5 \times 5 + 2.5 \times 10 + 5 \times 10)$

Step 4: Perform the calculations

Calculate each term:

$2.5 \times 5 = 12.5$

$2.5 \times 10 = 25$

$5 \times 10 = 50$

Now, calculate the total inside the parenthesis:

$A = 2(12.5 + 25 + 50) = 2(87.5)$

$A = 175 \text{ cm}^2$

Therefore, the surface area of the cuboid is $\mathbf{175 \text{ cm}^2}$.