Consider circle $Y$ with radius 3 m and central angle $X Y Z$ measuring $70^{\circ}$.
What is the approximate length of minor arc XZ?
Round to the nearest tenth of a meter.
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 (in radians): Arc length = 3 × 18 7 π = 6 7 π .
Approximate the value using π ≈ 3.14159 : Arc length ≈ 6 7 × 3.14159 ≈ 3.665188 .
Round to the nearest tenth of a meter: 3.7 meters .
Explanation
Problem Analysis We are given a circle with radius 3 m and a central angle of 70 degrees. We need to find the length of the minor arc XZ.
Convert angle 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 = 3 × 18 7 π = 18 21 π = 6 7 π meters
Approximate and round Now, we approximate the value to the nearest tenth of a meter. We know that π ≈ 3.14159 .
Arc length = 6 7 × 3.14159 ≈ 6 21.99113 ≈ 3.665188 meters Rounding to the nearest tenth of a meter, we get 3.7 meters.
Examples
Understanding arc length is crucial in many real-world applications. For instance, when designing a curved race track, engineers need to calculate the length of the curves to determine the total distance of the race. Similarly, in architecture, calculating the arc length of curved windows or doorways is essential for accurate material estimation and construction. Knowing how to find arc length allows for precise measurements and efficient planning in various fields.
