Circle O shown below has an arc of length 57 inches subtended by an angle of 106^(@). Find the length of the radius, x, to the nearest tenth of an inch. Answer: inches

Convert to Radians: To find the length of the radius, we can use the formula for the length of an arc, which is $ arc \ length = radius \times central \ angle \ (in \ radians) $. Since the central angle is given in degrees, we need to convert it to radians first.Calculate Angle: To convert degrees to radians, we use the conversion factor $ \frac{\pi \ radians}{180 \ degrees} $. So, the angle in radians is $ 106^{\circ} \times \frac{\pi}{180} $.Solve for Radius: Now, let's calculate the angle in radians: $ 106 \times \frac{\pi}{180} = \frac{106\pi}{180} $.Substitute Values: Next, we can rearrange the arc length formula to solve for the radius: $ radius = \frac{arc \ length}{central \ angle \ (in \ radians)} $.Perform Division: Substitute the given arc length and the calculated angle in radians into the formula: $ radius = \frac{57}{\frac{106\pi}{180}} $.Calculate Numerical Value: Now, perform the division to find the radius: $ radius = \frac{57 \times 180}{106\pi} $.Using a calculator to find the numerical value of the radius: $ radius \approx \frac{57 \times 180}{106 \times \pi} \approx \frac{10260}{333.038} \approx 30.8 $ inches.

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Circle
O shown below has an arc of length 57 inches subtended by an angle of
106^(@).
Find the length of the radius,
x, to the nearest tenth of an inch.
Answer: inches

Circle $O$ shown below has an arc of length $57$ inches subtended by an angle of $106^{\circ}$ . $\newline$ Find the length of the radius, $x$ , to the nearest tenth of an inch. $\newline$ Answer: $\square$ inches

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Grade 8

Circles: word problems

Full solution

Q. Circle

O

shown below has an arc of length

57

inches subtended by an angle of

106^{\circ}

\newline

Find the length of the radius,

x

, to the nearest tenth of an inch.

\newline

Answer:

\square

inches

Convert to Radians: To find the length of the radius, we can use the formula for the length of an arc, which is $arc \ length = radius \times central \ angle \ (in \ radians)$ . Since the central angle is given in degrees, we need to convert it to radians first.
Calculate Angle: To convert degrees to radians, we use the conversion factor $\frac{\pi \ radians}{180 \ degrees}$ . So, the angle in radians is $106^{\circ} \times \frac{\pi}{180}$ .
Solve for Radius: Now, let's calculate the angle in radians: $106 \times \frac{\pi}{180} = \frac{106\pi}{180}$ .
Substitute Values: Next, we can rearrange the arc length formula to solve for the radius: $radius = \frac{arc \ length}{central \ angle \ (in \ radians)}$ .
Perform Division: Substitute the given arc length and the calculated angle in radians into the formula: $radius = \frac{57}{\frac{106\pi}{180}}$ .
Calculate Numerical Value: Now, perform the division to find the radius: $radius = \frac{57 \times 180}{106\pi}$ .
Calculate Numerical Value: Now, perform the division to find the radius: $radius = \frac{57 \times 180}{106\pi}$ .Using a calculator to find the numerical value of the radius: $radius \approx \frac{57 \times 180}{106 \times \pi} \approx \frac{10260}{333.038} \approx 30.8$ inches.