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A circle has a circumference of
4pi feet (ft). An arc,
x, in this circle has a central angle of
240^(@). What is the length of
x ?
Choose 1 answer:
(A)
(4pi)/(3)ft
(B)
(8pi)/(3)ft
(C)
480ft
(D)
960ft

A circle has a circumference of $4 \pi$ feet (ft). An arc, $x$ , in this circle has a central angle of $240^{\circ}$ . What is the length of $x$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{4 \pi}{3} \mathrm{ft}$ $\newline$ (B) $\frac{8 \pi}{3} \mathrm{ft}$ $\newline$ (C) $480 \mathrm{ft}$ $\newline$ (D) $960 \mathrm{ft}$

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Scale drawings: word problems

Full solution

Q. A circle has a circumference of

4 \pi

feet (ft). An arc,

x

, in this circle has a central angle of

240^{\circ}

. What is the length of

x

?

\newline

Choose

1

answer:

\newline

(A)

\frac{4 \pi}{3} \mathrm{ft}

\newline

(B)

\frac{8 \pi}{3} \mathrm{ft}

\newline

(C)

480 \mathrm{ft}

\newline

(D)

960 \mathrm{ft}

Circumference Calculation: The circumference $C$ of a circle is related to its radius $r$ by the formula $C = 2\pi r$ . We are given that the circumference is $4\pi$ ft, so we can solve for $r$ : $\newline$ $4\pi = 2\pi r$ $\newline$ $r = 2$ ft
Central Angle Conversion: The length of an arc $s$ in a circle is given by the formula $s = r\theta$ , where $\theta$ is the central angle in radians. To use this formula, we need to convert the central angle from degrees to radians. The conversion factor is $\pi$ radians = $180^\circ$ , so: $\newline$ $\theta = 240^\circ \times (\pi/180^\circ) = (4/3)\pi$ radians
Arc Length Calculation: Now we can calculate the length of the arc $x$ using the radius we found in step $1$ and the angle in radians from step $2$ : $s = r\theta = 2 \, \text{ft} \times \left(\frac{4}{3}\right)\pi = \left(\frac{8}{3}\right)\pi \, \text{ft}$

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