In order to determine the appropriate length for the longest side of an acute triangle with two pieces of wood already cut to lengths of 7 inches and 3 inches, we need to explore the properties of acute triangles.
For a triangle to be acute, the square of the length of the longest side must be less than the sum of the squares of the other two sides. Let's breakdown the steps step by step:
1. Identify the given lengths of the wood pieces:
- [tex]\( a = 7 \)[/tex] inches
- [tex]\( b = 3 \)[/tex] inches
2. Calculate the squares of these lengths:
- [tex]\( a^2 = 7^2 = 49 \)[/tex]
- [tex]\( b^2 = 3^2 = 9 \)[/tex]
3. Determine the sum of the squares of these two lengths:
- [tex]\( a^2 + b^2 = 49 + 9 = 58 \)[/tex]
4. Find the length such that the triangle remains acute:
- The condition for a triangle to be acute is that the square of the length of the longest side, [tex]\( c \)[/tex], must be less than the sum of the squares of the other two sides:
[tex]\[ c^2 < a^2 + b^2 \][/tex]
- So, [tex]\( c^2 < 58 \)[/tex]
5. Calculate the maximum length [tex]\( c \)[/tex] by taking the square root of 58:
- [tex]\( c < \sqrt{58} \)[/tex]
- Numerically, [tex]\( \sqrt{58} \approx 7.615 \)[/tex]
Therefore, for Ramon to make an acute triangle with pieces of wood of lengths 7 inches and 3 inches, the longest side must be:
- Less than [tex]\( \sqrt{58} \)[/tex] inches but greater than 7 inches (since it's the longest side);
Evaluating the provided choices:
- Greater than [tex]\( \sqrt{58} \)[/tex] inches but less than 10 inches does not form an acute triangle, since the longest side would exceed the required limit.
- Exactly [tex]\( \sqrt{58} \)[/tex] inches, while numerically accurate as an upper bound, does not strictly satisfy [tex]\( c < \sqrt{58} \)[/tex] for all purposes of calculation.
- Less than [tex]\( \sqrt{58} \)[/tex] inches but greater than 7 inches is indeed the right solution, ensuring we adhere to the condition of an acute triangle.
Ultimately, the correct choice is:
"Less than [tex]\( \sqrt{58} \)[/tex] inches but greater than 7 inches"
