To solve this problem, we need to determine a few key expressions related to the square concert stage.
### Step-by-Step Solution
1. Represent the Area Expression:
We are given the area of the square stage as:
[tex]\[
4x^2 + 12x + 9 \text{ square feet}
\][/tex]
2. Identify the Side Length of the Square:
For a square, the area [tex]\(A\)[/tex] is given by the side length squared. Hence, we need to express the given area expression in the form [tex]\( (cx + d)^2 \)[/tex].
Notice that:
[tex]\[
(2x + 3)^2 = (2x + 3)(2x + 3) = 4x^2 + 12x + 9
\][/tex]
Therefore, the side length of the square stage is:
[tex]\[
2x + 3 \text{ feet}
\][/tex]
3. Find the Perimeter of the Square:
The perimeter [tex]\(P\)[/tex] of a square is four times the side length. Thus:
[tex]\[
P = 4 \times \text{side length} = 4 \times (2x + 3)
\][/tex]
4. Simplify the Perimeter Expression:
Simplifying this we get:
[tex]\[
P = 4 \times (2x + 3) = 8x + 12 \text{ feet}
\][/tex]
5. Determine the Perimeter for [tex]\(x = 2\)[/tex]:
We substitute [tex]\(x = 2\)[/tex] into the perimeter expression:
[tex]\[
P = 8(2) + 12 = 16 + 12 = 28 \text{ feet}
\][/tex]
### Summary
- Expression for the Perimeter:
[tex]\[
8x + 12 \text{ feet}
\][/tex]
- Perimeter when [tex]\(x = 2\)[/tex]:
[tex]\[
28 \text{ feet}
\][/tex]
Thus, the perimeter of the square concert stage when [tex]\(x = 2\)[/tex] feet is 28 feet.
