Answer :
To determine the smallest possible perimeter of an acute triangle with the longest side measuring 30 inches and the two remaining sides being congruent, we follow these steps:
1. Let the length of each congruent side be denoted as [tex]\( x \)[/tex].
2. Since the triangle is acute, it must satisfy the property of acute triangles: the square of the longest side must be less than the sum of the squares of the other two sides. Mathematically, this is expressed as:
[tex]\[ (30)^2 < 2x^2 \][/tex]
3. Simplify the inequality:
[tex]\[ 900 < 2x^2 \][/tex]
4. Divide both sides of the inequality by 2:
[tex]\[ 450 < x^2 \][/tex]
5. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x > \sqrt{450} \][/tex]
6. Compute [tex]\( \sqrt{450} \)[/tex], which is approximately [tex]\( 21.213203435596427 \)[/tex].
7. The smallest possible value for the congruent sides is hence [tex]\( x \approx 21.213 \)[/tex].
8. The perimeter of the triangle is the sum of all its sides. Therefore, the perimeter is:
[tex]\[ 30 + 2x \][/tex]
Substituting the computed value of [tex]\( x \)[/tex] into the equation:
[tex]\[ 30 + 2 \times 21.213203435596427 \approx 30 + 42.42640687119285 \approx 72.42640687119285 \][/tex]
9. Finally, round the perimeter to the nearest tenth:
[tex]\[ 72.42640687119285 \approx 72.4 \][/tex]
Thus, the smallest possible perimeter of the triangle, rounded to the nearest tenth, is [tex]\( 72.4 \)[/tex] inches.
Therefore, the correct answer is [tex]\( 72.4 \)[/tex] inches.
1. Let the length of each congruent side be denoted as [tex]\( x \)[/tex].
2. Since the triangle is acute, it must satisfy the property of acute triangles: the square of the longest side must be less than the sum of the squares of the other two sides. Mathematically, this is expressed as:
[tex]\[ (30)^2 < 2x^2 \][/tex]
3. Simplify the inequality:
[tex]\[ 900 < 2x^2 \][/tex]
4. Divide both sides of the inequality by 2:
[tex]\[ 450 < x^2 \][/tex]
5. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x > \sqrt{450} \][/tex]
6. Compute [tex]\( \sqrt{450} \)[/tex], which is approximately [tex]\( 21.213203435596427 \)[/tex].
7. The smallest possible value for the congruent sides is hence [tex]\( x \approx 21.213 \)[/tex].
8. The perimeter of the triangle is the sum of all its sides. Therefore, the perimeter is:
[tex]\[ 30 + 2x \][/tex]
Substituting the computed value of [tex]\( x \)[/tex] into the equation:
[tex]\[ 30 + 2 \times 21.213203435596427 \approx 30 + 42.42640687119285 \approx 72.42640687119285 \][/tex]
9. Finally, round the perimeter to the nearest tenth:
[tex]\[ 72.42640687119285 \approx 72.4 \][/tex]
Thus, the smallest possible perimeter of the triangle, rounded to the nearest tenth, is [tex]\( 72.4 \)[/tex] inches.
Therefore, the correct answer is [tex]\( 72.4 \)[/tex] inches.
