Question
W. W. Arc length 729 The radius of a circle is 8 miles. What is the length of a \( 90^{\circ} \) arc? Give the exact answer in simplest form. \( \square \) miles
Ask by Bryant Dickson. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The length of the \( 90^{\circ} \) arc is \( 4\pi \) miles.
Solution
We are given a circle with radius \( r = 8 \) miles and an arc defined by an angle of \( 90^{\circ} \). The arc length \( s \) of a circle is given by the formula:
\[
s = \left(\frac{\theta}{360^{\circ}}\right) \times (2\pi r)
\]
Substitute the given values:
1. Calculate the circumference of the circle:
\[
2\pi r = 2\pi(8) = 16\pi
\]
2. Compute the fraction of the circle corresponding to \( 90^{\circ} \):
\[
\frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}
\]
3. Multiply the circumference by this fraction to get the arc length:
\[
s = \frac{1}{4}(16\pi) = 4\pi
\]
Thus, the exact length of the \( 90^{\circ} \) arc is \( \boxed{4\pi} \) miles.
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Simplify this solution The Deep Dive
To find the length of a \( 90^{\circ} \) arc, you can use the formula for arc length, which is given by \( L = r \cdot \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. First, convert \( 90^{\circ} \) to radians: \( 90^{\circ} = \frac{\pi}{2} \) radians. Now plug in the values: \[ L = 8 \text{ miles} \cdot \frac{\pi}{2} = 4\pi \text{ miles} \] The exact length of the \( 90^{\circ} \) arc is \( 4\pi \) miles.
