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How long is the arc intersected by a central angle of [tex]$\frac{\pi}{3}$[/tex] radians in a circle with a radius of 6 ft? Round your answer to the nearest tenth. Use 3.14 for [tex]$\pi$[/tex].

A. 1.0 ft
B. 5.7 ft
C. 6.3 ft
D. 7.0 ft


Sagot :

To determine the length of an arc intercepted by a central angle in a circle, we use the formula for arc length:

[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle in Radians} \][/tex]

Given:
- Radius ([tex]\(r\)[/tex]) = 6 feet
- Central Angle ([tex]\(\theta\)[/tex]) = [tex]\(\frac{\pi}{3}\)[/tex] radians
- Use [tex]\(\pi \approx 3.14\)[/tex]

First, we substitute the given values into the formula:

[tex]\[ \text{Arc Length} = 6 \times \left(\frac{3.14}{3}\right) \][/tex]

Next, simplify the expression inside the parentheses:

[tex]\[ \frac{3.14}{3} \approx 1.0466667 \][/tex]

Now, multiply the radius by the result:

[tex]\[ \text{Arc Length} = 6 \times 1.0466667 \approx 6.279999999999999 \][/tex]

After finding the arc length, we round it to the nearest tenth:

[tex]\[ 6.279999999999999 \approx 6.3 \][/tex]

Therefore, the arc length intersected by a central angle of [tex]\(\frac{\pi}{3}\)[/tex] radians in a circle with a radius of 6 feet, rounded to the nearest tenth, is:

[tex]\[ \boxed{6.3 \text{ ft}} \][/tex]