In a circle, the minor arc AB has a central angle of 30 degrees and radii of 20 cm.

Find the length of the major arc AB correct to one decimal place.

Question

High School

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Answer :

SophiaElizab SophiaElizab · Answer 1 · 2024-04-15T11:06:42-04:00

In this problem, we are dealing with arcs of a circle. The circle has a known central angle for the minor arc AB and a given radius.

Understand the Circle's Full Angle:
- A circle has a total of 360 degrees.
Determine the Central Angle for the Major Arc:
- The central angle for the minor arc AB is given as 30 degrees.
- The major arc is the rest of the circle, so its angle is:
  [tex]360^\circ - 30^\circ = 330^\circ[/tex]
Find the Length of the Major Arc:
- The formula for the arc length [tex]L[/tex] is:
  [tex]L = \frac{\text{Central Angle in degrees}}{360} \times 2 \pi r[/tex]
- Plug in the values:
  [tex]L = \frac{330}{360} \times 2 \pi \times 20[/tex]
Perform the Calculations:
- Simplify the fraction:
  [tex]\frac{330}{360} = \frac{11}{12}[/tex]
- Therefore:
  [tex]L = \frac{11}{12} \times 2 \pi \times 20[/tex]
- Calculate:
  [tex]L = \frac{11}{6} \pi \times 20 = \frac{220}{6} \pi[/tex]
- Using [tex]\pi \approx 3.14159[/tex]:
  [tex]L \approx \frac{220}{6} \times 3.14159 \approx 115.2 \text{ cm}[/tex]
Final Answer:
- The length of the major arc AB, correct to one decimal place, is approximately 115.2 cm.