Surface Area 4x² + 24x: Finding Rectangular Prism Height

To solve the problem, we utilize the surface area formula for a rectangular prism, where the given prism has a base of dimensions $x$ and $2x$, and an unknown height $h$. The full formula for surface area is given as:

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

In this situation, $l = x$, $w = 2x$, and $h$ is our unknown. Substituting these values, the formula becomes:

$A = 2(x)(2x) + 2(x)h + 2(2x)h$

This simplifies to:

$A = 4x^2 + 2xh + 4xh$

Further simplification gives:

$A = 4x^2 + 6xh$

We are given the total surface area as $4x^2 + 24x$. Setting this equal to our expression:

$4x^2 + 6xh = 4x^2 + 24x$

Subtract $4x^2$ from both sides:

$6xh = 24x$

We can then divide both sides by $6x$ to solve for $h$:

$h = \frac{24x}{6x}$

This simplifies to:

$h = 4$

Thus, the height of the rectangular prism is $4$ cm.