To find the perimeter of a triangle, we need to sum the lengths of its sides. Given the side lengths of [tex]\(4 x^2 - 3\)[/tex] inches, [tex]\(4 x^2 - 2\)[/tex] inches, and [tex]\(4 x^2 - 1\)[/tex] inches, let's add them together to form a single expression for the perimeter.
1. Sum the expressions for the side lengths:
[tex]\[
(4 x^2 - 3) + (4 x^2 - 2) + (4 x^2 - 1)
\][/tex]
2. Combine like terms:
[tex]\[
4 x^2 + 4 x^2 + 4 x^2 - 3 - 2 - 1
\][/tex]
3. Simplify the expression:
[tex]\[
12 x^2 - 6
\][/tex]
So, the expression for the perimeter of the triangle is [tex]\(12 x^2 - 6\)[/tex].
Next, we need to find the perimeter when [tex]\(x = 1.5\)[/tex].
4. Substitute [tex]\(x = 1.5\)[/tex] into the perimeter expression:
[tex]\[
12 (1.5)^2 - 6
\][/tex]
5. Compute the squared term:
[tex]\[
(1.5)^2 = 2.25
\][/tex]
6. Multiply by 12:
[tex]\[
12 \cdot 2.25 = 27
\][/tex]
7. Subtract 6:
[tex]\[
27 - 6 = 21
\][/tex]
Thus, the expression for the perimeter of the triangle is [tex]\(12 x^2 - 6\)[/tex], and the perimeter when [tex]\(x = 1.5\)[/tex] is 21 inches.
Therefore, the correct answer is:
[tex]\[
12 x^2 - 6 ; 21 \text{ inches}
\][/tex]
