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Chapter 15: Problem 278

In a circle whose radius is 8 inches, find the number of degrees contained in the central angle whose arc length is \(2 \pi\) inches.

Short Answer

Expert verified
The central angle whose arc length is \(2\pi\) inches in a circle with radius 8 inches is 45掳.

Step by step solution

01

Write down the given information.

We are given the radius (r) of the circle, which is 8 inches, and the arc length (L), which is \(2\pi\) inches.
02

Use the formula for arc length to find the angle in radians.

The formula for arc length is: $$L = r\cdot \theta$$ where L is the arc length, r is the radius, and 胃 is the angle in radians. We are given L and r, so we can solve for 胃: $$\theta = \frac{L}{r}$$
03

Substitute the given values into the formula.

We are given L = \(2\pi\) inches and r = 8 inches. Substitute these values into the formula: $$\theta = \frac{2\pi}{8}$$
04

Simplify the expression.

Divide the constant in the numerator (2) by the constant in the denominator (8), and simplify the expression: $$\theta = \frac{\pi}{4}$$
05

Convert radians to degrees.

To convert radians to degrees, we can use the following conversion factor: $$1\, rad = \frac{180}{\pi}掳$$ And the conversion equation: $$Degrees = Radians \times \frac{180}{\pi}$$ Now, plug in the value of 胃 we got in Step 4: $$Degrees = \frac{\pi}{4} \times \frac{180}{\pi}$$
06

Simplify the expression.

The 蟺 in the numerator and the 蟺 in the denominator cancel out, and we are left with: $$Degrees = \frac{1}{4} \times 180$$ $$Degrees = 45掳$$ So, the central angle (in degrees) whose arc length is \(2\pi\) inches is 45掳.

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