To solve the problem, let's first understand the given side lengths of the triangle. The side lengths are expressed in terms of [tex]\( x \)[/tex] and are given as:
1. [tex]\( 4x^2 - 3 \)[/tex] inches
2. [tex]\( 4x^2 - 2 \)[/tex] inches
3. [tex]\( 4x^2 - 1 \)[/tex] inches
### Step 1: Finding the Perimeter Expression
To find the expression for the perimeter of the triangle, we need to sum up the three side lengths.
The perimeter [tex]\( P \)[/tex] is given by:
[tex]\[ P = (4x^2 - 3) + (4x^2 - 2) + (4x^2 - 1) \][/tex]
Combining like terms:
[tex]\[ P = 4x^2 + 4x^2 + 4x^2 - 3 - 2 - 1 \][/tex]
[tex]\[ P = 12x^2 - 6 \][/tex]
Thus, the expression used to find the perimeter of the triangle is:
[tex]\[ 12x^2 - 6 \][/tex]
### Step 2: Calculating the Perimeter When [tex]\( x = 1.5 \)[/tex]
Next, we need to find the perimeter when [tex]\( x = 1.5 \)[/tex].
1. Calculate each side length:
- First side: [tex]\( 4(1.5)^2 - 3 \)[/tex]
- Second side: [tex]\( 4(1.5)^2 - 2 \)[/tex]
- Third side: [tex]\( 4(1.5)^2 - 1 \)[/tex]
First, calculate [tex]\( 1.5^2 \)[/tex]:
[tex]\[ 1.5^2 = 2.25 \][/tex]
Then, plug this value back into the expressions for each side:
- First side: [tex]\( 4(2.25) - 3 \)[/tex]
[tex]\[ 4 \times 2.25 - 3 = 9 - 3 = 6 \text{ inches} \][/tex]
- Second side: [tex]\( 4(2.25) - 2 \)[/tex]
[tex]\[ 4 \times 2.25 - 2 = 9 - 2 = 7 \text{ inches} \][/tex]
- Third side: [tex]\( 4(2.25) - 1 \)[/tex]
[tex]\[ 4 \times 2.25 - 1 = 9 - 1 = 8 \text{ inches} \][/tex]
Now, sum these side lengths to find the perimeter when [tex]\( x = 1.5 \)[/tex]:
[tex]\[ \text{Perimeter} = 6 + 7 + 8 = 21 \text{ inches} \][/tex]
So, the expression for the perimeter is [tex]\( 12x^2 - 6 \)[/tex] and the perimeter when [tex]\( x = 1.5 \)[/tex] is [tex]\( 21 \)[/tex] inches. Therefore, the correct answer is:
[tex]\[ \boxed{12 x^2-6; 21 \text{ inches}} \][/tex]
