For questions 13 and 14, find the length 13. 232° 5 mm
The Correct Answer and Explanation is:
Of course. Here is the solution presented in a detailed, textbook-style explanation.
Solution for Question 13: Finding the Length of an Arc
Objective: To calculate the length of the major arc of a circle with a given radius and central angle.
Introduction:
An arc is a portion of the circumference of a circle. The length of an arc is determined by the size of its central angle and the radius of the circle. The problem provides a circle with a radius of 5 mm and indicates a major arc defined by a central angle of 232°. The task is to find the length of this specific arc.
Principle and Formula:
The length of an arc is a fraction of the total circumference of the circle. This fraction is equal to the ratio of the arc’s central angle (θ) to the total angle in a circle (360°).
The formula for the circumference of a circle is:
- C = 2πr
Combining these, the formula for the length of an arc (L) is:
- L = (θ / 360°) × 2πr
Step-by-Step Calculation:
- Identify the given values:
- Radius (r) = 5 mm
- Central Angle (θ) = 232°
- Substitute the values into the arc length formula:
- L = (232 / 360) × 2π(5)
- Simplify the expression:
- First, calculate the full circumference: C = 2π(5) = 10π mm.
- The formula becomes: L = (232 / 360) × 10π
- Reduce the fraction (232/360):
- To simplify the calculation, the fraction can be reduced by finding a common divisor. Both 232 and 360 are divisible by 8.
- 232 ÷ 8 = 29
- 360 ÷ 8 = 45
- The simplified fraction is 29/45.
- Calculate the exact arc length:
- Substitute the reduced fraction back into the equation: L = (29/45) × 10π
- Multiply the terms: L = (290π / 45)
- Simplify this final fraction by dividing the numerator and denominator by 5:
- L = (58π / 9) mm
This result, (58π/9) mm, is the exact length of the arc.
- Calculate the approximate decimal value:
- For a numerical approximation, use π ≈ 3.14159.
- L ≈ (58 / 9) × 3.14159
- L ≈ 6.444… × 3.14159
- L ≈ 20.246 mm
Final Answer:
The length of the arc subtended by the 232° angle is exactly (58π/9) mm. Rounding to two decimal places, the approximate length is 20.25 mm.thumb_upthumb_down
