To find the length of a major arc of a circle, we use the formula for arc length, which is:
[tex]\[ \text{Arc Length} = \left( \frac{\text{Central Angle}}{360} \right) \times 2 \pi \times \text{Radius} \][/tex]
Given:
- The radius of the circle [tex]\(P\)[/tex] is 50 yards.
- The central angle for the major arc [tex]\(RST\)[/tex] is [tex]\(215^{\circ}\)[/tex].
Substitute the given values into the formula:
[tex]\[ \text{Arc Length} = \left( \frac{215}{360} \right) \times 2 \pi \times 50 \][/tex]
First, simplify the fraction and multiply by [tex]\(2 \pi\)[/tex] and the radius:
[tex]\[ \text{Arc Length} = \left( \frac{215}{360} \right) \times 2 \pi \times 50 \][/tex]
[tex]\[ = \left( \frac{215}{360} \right) \times 100 \pi \][/tex]
Now compute the numerical value of the fraction:
[tex]\[ \frac{215}{360} = \frac{43}{72} \][/tex]
Next, multiply this fraction by [tex]\(100 \pi\)[/tex]:
[tex]\[ \left( \frac{43}{72} \right) \times 100 \pi = \frac{4300}{72} \pi \][/tex]
[tex]\[ \frac{4300}{72} = \frac{1075}{18} \][/tex]
Therefore, the arc length of major arc RST is:
[tex]\[ \frac{1075}{18} \pi \text{ yards} \][/tex]
The correct answer is:
[tex]\[ \boxed{\frac{1075}{18} \pi \text{ yards}} \][/tex]
