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Circle
O shown below has a radius of 11 inches. Find, to the nearest tenth of
a degree, the measure of the angle,
x, that forms an arc whose length is 24 inches.
Answer:

Circle $O$ shown below has a radius of $11$ inches. Find, to the nearest tenth of $a$ degree, the measure of the angle, $x$ , that forms an arc whose length is $24$ inches. $\newline$ Answer: $\square^{\circ}$

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Circles: word problems

Full solution

Q. Circle

O

shown below has a radius of

11

inches. Find, to the nearest tenth of

a

degree, the measure of the angle,

x

, that forms an arc whose length is

24

inches.

\newline

Answer:

\square^{\circ}

Understand Relationship: Understand the relationship between the radius, the arc length, and the central angle in a circle. $\newline$ The arc length $L$ of a circle is related to the central angle $\theta$ , in radians, and the radius $r$ by the formula $L = r\theta$ . To find the angle in degrees, we will need to convert from radians to degrees using the conversion factor $\frac{180}{\pi}$ .
Express Angle in Radians: Use the formula to express the angle in radians. $\newline$ We have the arc length $L = 24$ inches and the radius $r = 11$ inches. Plugging these values into the formula gives us $24 = 11\theta$ .
Solve for $\theta$ : Solve for $\theta$ in radians. $\newline$ Divide both sides of the equation by $11$ to isolate $\theta$ . $\newline$ $\theta = \frac{24}{11}$ $\newline$ $\theta \approx 2.1818$ radians
Convert to Degrees: Convert the angle from radians to degrees. $\newline$ To convert radians to degrees, multiply by $180/\pi$ . $\newline$ $\theta$ in degrees = $2.1818 \times (180/\pi)$ $\newline$ $\theta$ in degrees $\approx 2.1818 \times 57.2958$ $\newline$ $\theta$ in degrees $\approx 125.016$ degrees
Round to Nearest Tenth: Round the angle to the nearest tenth of a degree. $\theta$ in degrees $\approx 125.0$ degrees

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\newline

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