Consider circle [tex]$Y$[/tex] with radius 3 m and central angle [tex]$X Y Z$[/tex] measuring [tex]$70^{\circ}$[/tex]. What is the approximate length of minor arc [tex]$X Z$[/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 (1)
Convert the central angle from degrees to radians: 7 0 ∘ × 18 0 ∘ π = 18 7 π .
Calculate the arc length using the formula: arc length = radius × central angle. Arc length = 3 × 18 7 π = 6 7 π .
Approximate the value: 6 7 π ≈ 6 7 × 3.14159 ≈ 3.7 .
The approximate length of minor arc XZ is: 3.7 meters .
Explanation
Problem Analysis We are given a circle with radius r = 3 m and a central angle X Y Z = 7 0 ∘ . We need to find the length of the minor arc XZ .
Convert to Radians First, we need to convert the central angle from degrees to radians. To do this, we use the conversion factor 18 0 ∘ π :
7 0 ∘ × 18 0 ∘ π = 180 70 π = 18 7 π radians
Calculate Arc Length Next, we calculate the arc length using the formula: arc length = radius × central angle (in radians). Arc length = r × θ = 3 × 18 7 π = 18 21 π = 6 7 π m
Approximate and Round Now, we approximate the value of 6 7 π to the nearest tenth of a meter. Using the approximation π ≈ 3.14159 , we have: 6 7 π ≈ 6 7 × 3.14159 ≈ 6 21.99113 ≈ 3.665188 ≈ 3.7 m
Final Answer Therefore, the approximate length of minor arc XZ is 3.7 meters.
Examples
Imagine you're designing a curved pathway around a circular garden. Knowing the radius of the garden and the angle you want the pathway to cover, you can use the arc length formula to determine the exact length of paving stones you'll need. This ensures you purchase the right amount of materials, saving time and money. For instance, if the garden has a radius of 3 meters and you want the pathway to cover an angle of 70 degrees, you'll need approximately 3.7 meters of paving stones.
