Cuboid Dimensions: Solving for 3:5 Height-Width Ratio with 542 cm² Area

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

• Step 1: Express dimensions in terms of a single variable using the given ratios.

• Step 2: Substitute these expressions into the cuboid's surface area formula.

• Step 3: Solve for the variable and in turn calculate the dimensions.

Now, let's work through each step:

Step 1: Express dimensions using the ratios.
- Let $h = 3x$ (height is set from height to width ratio), $w = 5x$ (width), $l = \frac{7}{2}h = \frac{7}{2}(3x) = \frac{21}{2}x$ (length from 2:7 as basis relative to height).

Step 2: Substitute into the surface area formula.
The surface area $S$ is given by:
$S = 2(lw + lh + wh) = 542 \, \text{cm}^2$

Plugging in, we get:
$2\left(\frac{21}{2}x \cdot 5x + \frac{21}{2}x \cdot 3x + 3x \cdot 5x\right) = 542$

Simplifying further:
$2\left(\frac{105}{2}x^2 + \frac{63}{2}x^2 + 15x^2\right) = 542$

Combining terms, we get:
$2 \times \left(\frac{105x^2 + 63x^2 + 30x^2}{2}\right) = 542$

The combined denominator cancels out:
$198x^2 = 542$

Solving for $x^2$:
$x^2 = \frac{542}{198} \approx 2.737$

Solving for $x$:
$x \approx \sqrt{2.737} \approx 1.654$

Step 3: Calculate dimensions using $x$:
- $h = 3x \approx 3 \times 1.654 \approx 4.96 \, \text{cm}$
- $w = 5x \approx 5 \times 1.654 \approx 8.26 \, \text{cm}$
- $l = \frac{21}{2}x \approx \frac{21}{2} \times 1.654 \approx 17.36 \, \text{cm}$

Therefore, the solution to the problem is $\text{Height } = 4.96 \, \text{cm}, \text{ Length } = 17.36 \, \text{cm}, \text{ Width } = 8.26 \, \text{cm}$, which matches choice 4.