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A circle has a circumference of 
12 pi feet (ft). An arc, 
x, in this circle has a central angle of 
330^(@). Rounded to the nearest foot, what is the length of 
x ?

A circle has a circumference of 12π 12 \pi feet (ft) (\mathrm{ft}) . An arc, x x , in this circle has a central angle of 330 330^{\circ} . Rounded to the nearest foot, what is the length of x x ?

Full solution

Q. A circle has a circumference of 12π 12 \pi feet (ft) (\mathrm{ft}) . An arc, x x , in this circle has a central angle of 330 330^{\circ} . Rounded to the nearest foot, what is the length of x x ?
  1. Find Radius of Circle: Find the radius of the circle using the circumference.\newlineThe formula for the circumference of a circle is C=2πrC = 2 \pi r, where CC is the circumference and rr is the radius.\newlineGiven C=12πC = 12 \pi, we can solve for rr:\newline12π=2πr12 \pi = 2 \pi r\newlineDivide both sides by 2π2 \pi to isolate rr:\newliner=12π2πr = \frac{12 \pi}{2 \pi}\newliner=6r = 6
  2. Calculate Length of Arc: Calculate the length of the arc xx.\newlineThe formula for the length of an arc (ss) with a central angle (θ\theta) in degrees in a circle of radius rr is s=θ360×2×π×rs = \frac{\theta}{360} \times 2 \times \pi \times r.\newlineGiven θ=330\theta = 330 degrees and r=6r = 6 feet, we can plug these values into the formula:\newlines=330360×2×π×6s = \frac{330}{360} \times 2 \times \pi \times 6\newlines=1112×2×π×6s = \frac{11}{12} \times 2 \times \pi \times 6\newlines=11×π×612s = 11 \times \pi \times \frac{6}{12}\newliness00\newliness11
  3. Round Length of Arc: Round the length of the arc to the nearest foot.\newlineSince π\pi is approximately 3.141593.14159, we can estimate the length of the arc:\newlines5.5×3.14159s \approx 5.5 \times 3.14159\newlines17.278s \approx 17.278\newlineRounded to the nearest foot, the length of the arc xx is approximately 1717 feet.

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