QuestionFind the radius of a circle with a central angle of $30^{\circ}$ that intercepts an arc of length $17 \mathrm{mi}$ . Radius in terms of $\pi$ .

Studdy Solution

STEP 1

Assumptions1. The central angle, $\theta$ , is $30^{\circ}$ . The arc length intercepted by this angle is $17$ mi3. We use the formula for the length of an arc $s = r\theta$ , where $s$ is the length of the arc, $r$ is the radius of the circle, and $\theta$ is the angle in radians.

STEP 2

First, we need to convert the central angle from degrees to radians because the formula for the length of an arc uses radians. We can do this using the conversion factor $\frac{\pi}{180}$ .
$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}$

STEP 3

Now, plug in the given value for the central angle to calculate the angle in radians.
$\theta_{radians} =30^{\circ} \times \frac{\pi}{180}$

STEP 4

Calculate the angle in radians.
$\theta_{radians} = \frac{\pi}{6}$

STEP 5

Now that we have the angle in radians, we can rearrange the formula for the length of an arc to solve for the radius.
$r = \frac{s}{\theta_{radians}}$

STEP 6

Plug in the values for the arc length and the angle in radians to calculate the radius.
$r = \frac{17\, mi}{\frac{\pi}{6}}$

STEP 7

Calculate the radius of the circle.
$r = \frac{17\, mi \times6}{\pi} = \frac{102}{\pi}\, mi$ The radius of the circle is $\frac{102}{\pi}$ mi.

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QuestionFind the radius of a circle with a central angle of $30^{\circ}$ that intercepts an arc of length $17 \mathrm{mi}$ . Radius in terms of $\pi$ .

Studdy Solution

STEP 1

Assumptions1. The central angle, $\theta$ , is $30^{\circ}$ . The arc length intercepted by this angle is $17$ mi3. We use the formula for the length of an arc $s = r\theta$ , where $s$ is the length of the arc, $r$ is the radius of the circle, and $\theta$ is the angle in radians.

STEP 2

First, we need to convert the central angle from degrees to radians because the formula for the length of an arc uses radians. We can do this using the conversion factor $\frac{\pi}{180}$ .
$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}$

STEP 3

Now, plug in the given value for the central angle to calculate the angle in radians.
$\theta_{radians} =30^{\circ} \times \frac{\pi}{180}$

STEP 4

Calculate the angle in radians.
$\theta_{radians} = \frac{\pi}{6}$

STEP 5

Now that we have the angle in radians, we can rearrange the formula for the length of an arc to solve for the radius.
$r = \frac{s}{\theta_{radians}}$

STEP 6

Plug in the values for the arc length and the angle in radians to calculate the radius.
$r = \frac{17\, mi}{\frac{\pi}{6}}$

STEP 7

Calculate the radius of the circle.
$r = \frac{17\, mi \times6}{\pi} = \frac{102}{\pi}\, mi$ The radius of the circle is $\frac{102}{\pi}$ mi.

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QuestionFind the radius of a circle with a central angle of 30∘30^{\circ}30∘ that intercepts an arc of length 17mi17 \mathrm{mi}17mi. Radius in terms of π\piπ.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, plug in the given value for the central angle to calculate the angle in radians.θradians=30∘×π180\theta_{radians} =30^{\circ} \times \frac{\pi}{180}θradians​=30∘×180π​

STEP 4

Calculate the angle in radians.θradians=π6\theta_{radians} = \frac{\pi}{6}θradians​=6π​

STEP 5

Now that we have the angle in radians, we can rearrange the formula for the length of an arc to solve for the radius.r=sθradiansr = \frac{s}{\theta_{radians}}r=θradians​s​

STEP 6

Plug in the values for the arc length and the angle in radians to calculate the radius.r=17 miπ6r = \frac{17\, mi}{\frac{\pi}{6}}r=6π​17mi​

STEP 7

Calculate the radius of the circle.r=17 mi×6π=102π mir = \frac{17\, mi \times6}{\pi} = \frac{102}{\pi}\, mir=π17mi×6​=π102​miThe radius of the circle is 102π\frac{102}{\pi}π102​ mi.

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QuestionFind the radius of a circle with a central angle of $30^{\circ}$ that intercepts an arc of length $17 \mathrm{mi}$ . Radius in terms of $\pi$ .

Now, plug in the given value for the central angle to calculate the angle in radians.
$\theta_{radians} =30^{\circ} \times \frac{\pi}{180}$

Calculate the angle in radians.
$\theta_{radians} = \frac{\pi}{6}$

Now that we have the angle in radians, we can rearrange the formula for the length of an arc to solve for the radius.
$r = \frac{s}{\theta_{radians}}$

Plug in the values for the arc length and the angle in radians to calculate the radius.
$r = \frac{17\, mi}{\frac{\pi}{6}}$

Calculate the radius of the circle.
$r = \frac{17\, mi \times6}{\pi} = \frac{102}{\pi}\, mi$ The radius of the circle is $\frac{102}{\pi}$ mi.