Question

pablo folds a straw into a triangle with side lengths of $4x^2 - 3$ inches, $4x^2 - 2$ inches, and $4x^2 - 1$ inches. which expression can be used to find the perimeter of the triangle, and what is the perimeter when $x = 1.5$?
\\( 12x^2 - 6 \\); 21 inches
\\( 12x^2 \\); 27 inches
\\( 12x^2 - 6 \\); 30 inches
\\( 12x^2 \\); 36 inches

Explanation:

Step1: Find the perimeter expression

The perimeter \( P \) of a triangle is the sum of its three sides. The sides are \( 4x^2 - 3 \), \( 4x^2 - 2 \), and \( 4x^2 - 1 \). So we add them together:
\[

$$\begin{align*} P&=(4x^2 - 3)+(4x^2 - 2)+(4x^2 - 1)\\ &=4x^2+4x^2+4x^2-3 - 2 - 1\\ &=12x^2-6 \end{align*}$$

\]

Step2: Calculate perimeter when \( x = 1.5 \)

Substitute \( x = 1.5 \) into the expression \( 12x^2-6 \). First, calculate \( x^2=(1.5)^2 = 2.25 \). Then:
\[

$$\begin{align*} 12x^2-6&=12\times2.25-6\\ &=27 - 6\\ &=21 \end{align*}$$

\]
Wait, that's not matching the options? Wait, no, wait \( 1.5^2 = 2.25 \), \( 12\times2.25 = 27 \), \( 27-6 = 21 \)? But wait the options have 12x² -6 and 21? Wait no, let me check again. Wait, maybe I made a mistake. Wait \( 4x^2-3 + 4x^2 -2 +4x^2 -1=12x^2 -6 \). Then when \( x = 1.5 \), \( x^2 = 2.25 \), \( 12\times2.25 = 27 \), \( 27 -6 = 21 \). So the first option is \( 12x^2 -6 \); 21 inches. But wait the third option is 30? Wait maybe I miscalculated. Wait \( x = 1.5 \), \( x^2 = (3/2)^2 = 9/4 \). Then \( 12\times(9/4)=27 \), \( 27 -6 =21 \). So the correct option is the first one? Wait but let me check the options again. The options are:

1. \( 12x^2 -6 \); 21 inches

1. \( 12x^2 \); 27 inches

1. \( 12x^2 -6 \); 30 inches

1. \( 12x^2 \); 36 inches

Wait, maybe I made a mistake in the perimeter formula. Wait the sides are \( 4x^2 -3 \), \( 4x^2 -2 \), \( 4x^2 -1 \). So sum is \( 4x^2+4x^2+4x^2=12x^2 \), and \( -3-2-1=-6 \), so perimeter is \( 12x^2 -6 \). Then when \( x = 1.5 \), \( x^2 = 2.25 \), \( 12*2.25 = 27 \), \( 27 -6 =21 \). So the first option is correct. But wait the third option has 30. Maybe I misread the problem. Wait the problem says "4x²−3 inches, 4x²−2 inches, and 4x²−1 inches". Wait maybe the problem was 4x² +3? No, the user wrote 4x²−3. Wait let me check again. If x=2, then 4(4)=16, 16-3=13, 16-2=14, 16-1=15. Sum is 13+14+15=42. 12x² -6 when x=2 is 124 -6=48-6=42. Correct. When x=1.5, 12*(2.25)-6=27-6=21. So the first option is correct.

Answer:

A. \( 12x^2 - 6 \); 21 inches