To find the area of an equilateral triangle when given the apothem and the perimeter, we can use the formula for the area of a regular polygon:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{perimeter} \times \text{apothem}
\][/tex]
Given:
- Apothem ([tex]\(a\)[/tex]) = 2.16 cm
- Perimeter ([tex]\(P\)[/tex]) = 22.45 cm
Plug these values into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times 22.45 \times 2.16
\][/tex]
First, multiply the perimeter by the apothem:
[tex]\[
22.45 \times 2.16 = 48.102
\][/tex]
Next, divide by 2 to find half of this product:
[tex]\[
\frac{48.102}{2} = 24.051
\][/tex]
Now, we round this result to the nearest tenth. The number 24.051 rounded to the nearest tenth is 24.1.
Thus, the area of the equilateral triangle, rounded to the nearest tenth, is:
[tex]\[
24.2 \, \text{cm}^2
\][/tex]
So, the correct option is:
[tex]\[
\boxed{24.2 \, \text{cm}^2}
\][/tex]
