the base of a solid right pyramid is a regular hexagon with a radius of 2x units and an apothem of…

Explanation:

Step1: Recall area formula for regular polygon

The area formula for a regular polygon is $A = \frac{1}{2}ap$, where $a$ is the apothem and $p$ is the perimeter.

Step2: Find side - length of hexagon

For a regular hexagon with radius $r = 2x$, the side - length $s$ of the hexagon is equal to the radius, so $s=2x$.

Step3: Calculate perimeter of hexagon

The perimeter $p$ of a regular hexagon with side - length $s$ is $p = 6s$. Substituting $s = 2x$, we get $p=6\times2x = 12x$.

Step4: Calculate area of base

The apothem $a=x\sqrt{3}$. Using the area formula $A=\frac{1}{2}ap$, substitute $a = x\sqrt{3}$ and $p = 12x$. Then $A=\frac{1}{2}\times x\sqrt{3}\times12x=6x^{2}\sqrt{3}$ square units.

Answer:

$6x^{2}\sqrt{3}$ units$^{2}$