Select the correct answer.

A triangle has side lengths of [tex]n[/tex], [tex]n-3[/tex], and [tex]2(n-2)[/tex]. If the perimeter of the triangle is at least 37 units, what is the value of [tex]n[/tex]?

A. [tex]n \geq 10.5[/tex]
B. [tex]n \geq 11[/tex]
C. [tex]n \geq 8[/tex]
D. [tex]n \geq 7.5[/tex]



Answer :

To solve this problem, we need to determine the minimum value of \( n \) such that the perimeter of the triangle is at least 37 units.

Given the side lengths of the triangle:
- First side: \( n \)
- Second side: \( n - 3 \)
- Third side: \( 2(n - 2) \)

The perimeter \( P \) of the triangle is the sum of the lengths of its sides. Therefore, we can express the perimeter as:
[tex]\[ P = n + (n - 3) + 2(n - 2) \][/tex]

Simplify the expression:
[tex]\[ P = n + n - 3 + 2n - 4 \][/tex]
Combine like terms:
[tex]\[ P = 4n - 7 \][/tex]

We want the perimeter to be at least 37 units, so we set up the inequality:
[tex]\[ 4n - 7 \geq 37 \][/tex]

To solve for \( n \), first isolate \( n \) on one side of the inequality:
[tex]\[ 4n - 7 \geq 37 \][/tex]
Add 7 to both sides:
[tex]\[ 4n \geq 44 \][/tex]
Divide both sides by 4:
[tex]\[ n \geq 11 \][/tex]

Therefore, the minimum value of \( n \) that satisfies this condition is \( 11 \). Thus, the correct answer is:
B. [tex]\( n \geq 11 \)[/tex]