Select the correct answer.
Circle P has a radius of 50 yards. Major Arc RST on the circle has a central angle of [tex]$215^{\circ}$[/tex]. What is the length of major arc RST?
A. [tex]$\frac{1,075}{18} \pi$[/tex] yards
B. [tex]$\frac{1,075}{15} \pi$[/tex] yards
C. [tex]$130 \pi$[/tex] yards
D. [tex]$140 \pi$[/tex] yards
Answer (2)
Calculate the circumference of the circle: C = 2 ฯ r = 100 ฯ yards.
Determine the fraction of the circle represented by the major arc: 360 215 โ = 72 43 โ .
Multiply the circumference by the fraction to find the arc length: 72 43 โ ร 100 ฯ = 72 4300 โ ฯ .
Simplify the fraction to get the final answer: 18 1075 โ ฯ yards โ .
Explanation
Problem Analysis We are given a circle P with a radius of 50 yards. A major arc RST on the circle has a central angle of 21 5 โ . We need to find the length of major arc RST.
Calculate the Circumference First, we need to find the circumference of the circle. The formula for the circumference of a circle is given by: C = 2 ฯ r where r is the radius of the circle. In this case, the radius is 50 yards. So, the circumference is: C = 2 ฯ ( 50 ) = 100 ฯ yards
Determine the Fraction of the Circle Next, we need to find the fraction of the circle that the major arc RST represents. This is given by the ratio of the central angle of the arc to the total angle in a circle ( 36 0 โ ): Fraction = 36 0 โ Central Angle โ = 36 0 โ 21 5 โ โ We can simplify this fraction by dividing both the numerator and the denominator by 5: 360 215 โ = 360 รท 5 215 รท 5 โ = 72 43 โ
Calculate the Arc Length Now, we can find the length of the major arc RST by multiplying the circumference of the circle by the fraction we just found: Arc Length = Circumference ร Fraction = 100 ฯ ร 72 43 โ We can simplify this expression: Arc Length = 72 100 ร 43 โ ฯ = 72 4300 โ ฯ Now, we can simplify the fraction by dividing both the numerator and the denominator by 4: 72 4300 โ = 72 รท 4 4300 รท 4 โ = 18 1075 โ So, the length of the major arc RST is: Arc Length = 18 1075 โ ฯ yards
Final Answer The length of major arc RST is 18 1075 โ ฯ yards.
Examples
Imagine you're designing a circular track for a toy car. The track has a radius of 50 cm, and you want to place a special marking along the track that covers an arc of 215 degrees. To determine the exact length of the marking you'll need, you can use the same arc length calculation we just performed. This ensures the marking is precisely positioned to enhance the toy car's journey around the track.
The length of major arc RST on a circle with a radius of 50 yards and a central angle of 215 degrees is calculated to be 18 1075 โ ฯ yards. Therefore, the correct answer is option A. This is found by calculating the circumference, determining the fraction of the circle, and deriving the arc length accordingly.
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