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Circle
O shown below has a radius of 4 inches. Find, to the nearest tenth of a degree, the measure of the angle,
x, that forms an arc whose length is 7 inches.
Answer:

Circle $O$ shown below has a radius of $4$ inches. Find, to the nearest tenth of a degree, the measure of the angle, $x$ , that forms an arc whose length is $7$ inches. $\newline$ Answer: $\square^{\circ}$

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Circles: word problems

Full solution

Q. Circle

O

shown below has a radius of

4

inches. Find, to the nearest tenth of a degree, the measure of the angle,

x

, that forms an arc whose length is

7

inches.

\newline

Answer:

\square^{\circ}

Understand Arc Length Formula: To find the measure of angle x, we need to use the formula for the length of an arc, which is given by \text{Arc length} = 2\pi r \left(\frac{\theta}{360}\right) , where r is the radius of the circle and $\theta$ is the angle in degrees that subtends the arc at the center of the circle.
Identify Known Values: We know the radius (r) is $4$ inches and the arc length is $7$ inches. We need to solve for $\theta$ (the angle in degrees). Plugging in the known values, we get $7 = 2\pi(4) \left(\frac{\theta}{360}\right)$ .
Solve for Theta: Simplify the equation by multiplying both sides by $\frac{360}{2\pi(4)}$ to isolate $\theta$ . This gives us $\theta = \frac{7 \times 360}{2\pi(4)}$ .
Calculate Theta Value: Now, calculate the value of $\theta$ . $\theta = \frac{7 \times 360}{2\pi(4)} = \frac{7 \times 360}{8\pi}$ .
Round to Nearest Tenth: Using a calculator, we find that $\theta \approx \frac{7 \times 360}{8 \times 3.14159} \approx \frac{2520}{25.13272} \approx 100.24$ degrees.
Final Angle Measurement: Since we need to round to the nearest tenth of a degree, the measure of angle $x$ is approximately $100.2$ degrees.

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\newline

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\newline

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