To determine the length of the path Frankie wants to build, we need to calculate the diagonal distance from one corner of his yard to the opposite corner. This can be achieved using the Pythagorean theorem.
1. Understand the Problem:
- The yard is a rectangle with length [tex]\( L = 20 \)[/tex] feet and width [tex]\( W = 32 \)[/tex] feet.
- We need to find the diagonal distance [tex]\( D \)[/tex], which is the hypotenuse of the right triangle formed by the length and width.
2. Pythagorean Theorem:
- The Pythagorean theorem states that for a right triangle with legs [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and hypotenuse [tex]\( c \)[/tex]:
[tex]\[ c = \sqrt{a^2 + b^2} \][/tex]
- In our case:
[tex]\[ a = 20 \text{ ft} \][/tex]
[tex]\[ b = 32 \text{ ft} \][/tex]
[tex]\[ D = \sqrt{20^2 + 32^2} \][/tex]
3. Calculate the Squares:
- First, calculate the squares of the length and width:
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ 32^2 = 1024 \][/tex]
4. Sum the Squares:
- Add the squares together:
[tex]\[ 400 + 1024 = 1424 \][/tex]
5. Take the Square Root:
- Find the square root of the sum to get the length of the diagonal:
[tex]\[ D = \sqrt{1424} \approx 37.73592452822641 \][/tex]
6. Round to the Nearest Tenth:
- The diagonal distance, rounded to the nearest tenth of a foot, is approximately:
[tex]\[ 37.7 \text{ ft} \][/tex]
Hence, the length of Frankie’s path will be approximately 37.7 feet.
Therefore, the correct answer is:
- 37.7 ft
