Consider circle [tex]Y[/tex] with radius 3 m and central angle [tex]XYZ[/tex] measuring 70 degrees. What is the approximate length of minor arc [tex]XZ[/tex]? Round to the nearest tenth of a meter.
A. 1.8 meters
B. 3.7 meters
C. 15.2 meters
D. 18.8 meters
Answer (2)
Calculate the circumference of the circle using the formula C = 2 π r , where r = 3 m, resulting in C ≈ 18.85 m.
Determine the fraction of the circle represented by the 70-degree central angle: 360 70 = 36 7 .
Multiply the circumference by this fraction to find the arc length: 36 7 × 18.85 ≈ 3.665 m.
Round the arc length to the nearest tenth of a meter: 3.7 meters .
Explanation
Problem Analysis We are given a circle Y with radius 3 m and a central angle X Y Z measuring 70 degrees. Our goal is to find the length of the minor arc XZ , rounded to the nearest tenth of a meter.
Calculate the Circumference First, we need to calculate the circumference of the entire circle. The formula for the circumference C of a circle is given by: C = 2 π r where r is the radius of the circle. In this case, the radius r = 3 m. Substituting this value into the formula, we get: C = 2 π ( 3 ) = 6 π ≈ 18.85 m So, the circumference of the circle is approximately 18.85 meters.
Determine the Fraction of the Circle Next, we need to determine what fraction of the entire circle the minor arc XZ represents. The central angle X Y Z is 70 degrees, and a full circle is 360 degrees. Therefore, the fraction of the circle represented by the arc is: 360 70 = 36 7 This means the minor arc XZ is 36 7 of the entire circle.
Calculate the Arc Length Now, we can find the length of the minor arc XZ by multiplying the circumference of the circle by the fraction we just calculated: Arc Length = 36 7 × C = 36 7 × 6 π = 6 7 π ≈ 3.665 m So, the length of the minor arc XZ is approximately 3.665 meters.
Round to the Nearest Tenth Finally, we need to round the arc length to the nearest tenth of a meter. Since 3.665 is closer to 3.7 than to 3.6, we round up to 3.7 meters.
Therefore, the approximate length of the minor arc XZ is 3.7 meters.
Examples
Imagine you're designing a circular flower bed with a radius of 3 meters, and you want to plant a specific type of flower along a 70-degree arc of the circle's edge. Knowing how to calculate the arc length allows you to determine the exact length of the edge where you'll plant those flowers, ensuring you buy the right amount of plants. In this case, you'd need to plant flowers along approximately 3.7 meters of the circle's edge.
The length of minor arc XZ in a circle with a radius of 3 meters and a central angle of 70 degrees is approximately 3.7 meters. This is found by calculating the circumference of the circle, determining the fraction of the circle represented by the angle, and then finding the arc length. The correct multiple-choice answer is B. 3.7 meters .
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