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Circle
O shown below has a radius of 3 inches. Find, to the nearest tenth of a degree, the measure of the angle,
x, that forms an arc whose length is 4 inches.
Answer:

Circle $O$ shown below has a radius of $3$ inches. Find, to the nearest tenth of a degree, the measure of the angle, $x$ , that forms an arc whose length is $4$ inches. $\newline$ Answer: $\square^{\circ}$

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Circles: word problems

Full solution

Q. Circle

O

shown below has a radius of

3

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

x

, that forms an arc whose length is

4

inches.

\newline

Answer:

\square^{\circ}

Identify Formula: To find the measure of the angle x, we need to use the formula for the length of an arc, which is \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. $\newline$ Given: $\newline$ Radius (r) = $3$ inches $\newline$ Arc length = $4$ inches $\newline$ We need to solve for $\theta$ (the angle in degrees).
Given Values: First, let's plug in the given values into the arc length formula: $\newline$ $4 = 2\pi \times 3 \times \left(\frac{\theta}{360}\right)$
Plug in Values: Now, we simplify the equation: $\newline$ $4 = 6\pi \times \left(\frac{\theta}{360}\right)$
Simplify Equation: Next, we solve for $\theta$ by multiplying both sides of the equation by $\frac{360}{6\pi}$ : $\newline$ $\theta = \frac{4 \times 360}{6\pi}$
Solve for Theta: Now, we calculate the value of $\theta$ : $\newline$ $\theta = \frac{4 \times 360}{6 \times \pi}$ $\newline$ $\theta = \frac{1440}{6\pi}$ $\newline$ $\theta = \frac{240}{\pi}$
Calculate Theta: Finally, we use a calculator to find the numerical value of $\theta$ to the nearest tenth of a degree: $\newline$ $\theta \approx \frac{240}{3.14159}$ $\newline$ $\theta \approx 76.4$ degrees

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\newline

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