Let's solve this problem step-by-step.
1. Understanding the Problem:
We have a triangular region where all sides are of the same length, hence it is an equilateral triangle. The perimeter of this equilateral triangle is given as 90 feet.
2. Finding the Side Length:
In an equilateral triangle, all three sides are equal. Therefore, the side length can be found by dividing the perimeter by 3:
[tex]\[
\text{Side length} = \frac{\text{Perimeter}}{3} = \frac{90}{3} = 30 \, \text{feet}
\][/tex]
3. Calculating the Area:
The formula for the area of an equilateral triangle with side length [tex]\( a \)[/tex] is:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times a^2
\][/tex]
Plugging in the side length [tex]\( a = 30 \)[/tex] feet, we get:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times (30)^2 = \frac{\sqrt{3}}{4} \times 900
\][/tex]
4. Simplifying the Expression:
We now perform the multiplication:
[tex]\[
\text{Area} \approx \frac{1.732}{4} \times 900
\][/tex]
Simplifying further:
[tex]\[
\frac{1.732}{4} \approx 0.433
\][/tex]
[tex]\[
0.433 \times 900 \approx 389.7 \, \text{square feet}
\][/tex]
5. Rounding to the Nearest Square Foot:
We need to round the area to the nearest square foot:
[tex]\[
389.7 \approx 390 \, \text{square feet}
\][/tex]
So, the area that is reserved for the children's chalk art is approximately [tex]\( 390 \, \text{square feet} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{390 \, \text{ft}^2}
\][/tex]
