17. an angle is 4.11 radians and the length of an arc subtended by the angle is 26. what is the radius of the circle?
18. radius is 7.8 and the length of an arc subtended by an angle is 17.94. find the measure of the angle.
19. an angle is 3.06 radians and the length of an arc subtended by the angle is 29.47. what is the radius of the circle?
3.2a radians
3.2a test prep
20. an angle with a measure of 5.7π would be equivalent to which of the following angle measures?
(a) 2.7π
(b) −1.7π
(c) −5.7π
(d) −0.3π

Question

Accepted Answer

### Question 17
# Explanation:
## Step1: Recall the arc length formula $ s = r\theta $ (where $ s $ is arc length, $ r $ is radius, $ \theta $ is angle in radians)
We know $ s = 26 $ and $ \theta = 4.11 $ radians. We need to solve for $ r $, so rearrange the formula: $ r=\frac{s}{\theta} $
## Step2: Substitute the values into the formula
$ r = \frac{26}{4.11} \approx 6.33 $
# Answer:
The radius of the circle is approximately $ 6.33 $

### Question 18
# Explanation:
## Step1: Recall the arc length formula $ s = r\theta $, solve for $ \theta $: $ \theta=\frac{s}{r} $
We know $ s = 17.94 $ and $ r = 7.8 $
## Step2: Substitute the values into the formula
$ \theta=\frac{17.94}{7.8}=2.3 $ radians
# Answer:
The measure of the angle is $ 2.3 $ radians

### Question 19
# Explanation:
## Step1: Recall the arc length formula $ s = r\theta $, solve for $ r $: $ r=\frac{s}{\theta} $
We know $ s = 29.47 $ and $ \theta = 3.06 $ radians
## Step2: Substitute the values into the formula
$ r=\frac{29.47}{3.06}\approx9.63 $
# Answer:
The radius of the circle is approximately $ 9.63 $

### Question 20
# Explanation:
To find an equivalent angle, we can use the concept that adding or subtracting $ 2\pi $ (or $ 4\pi $, $ 6\pi $, etc., since we are dealing with $ \pi $ multiples here) to an angle gives a coterminal angle. Let's see how many times $ 2\pi $ (or $ 4\pi $) we can subtract from $ 5.7\pi $ to get an equivalent angle within a reasonable range.

First, note that angles that differ by $ 2\pi $ (or $ 4\pi $, $ 6\pi $, etc.) are coterminal. Let's subtract $ 4\pi $ from $ 5.7\pi $: $ 5.7\pi - 4\pi = 1.7\pi $. But this is not an option. Wait, maybe we consider negative coterminal angles. Let's subtract $ 6\pi $ from $ 5.7\pi $: $ 5.7\pi - 6\pi=- 0.3\pi $. Wait, no, let's check the options. Wait, another approach: equivalent angles can also be found by adding or subtracting $ 2k\pi $, but since we are dealing with $ \pi $ as the base (the options are in terms of $ \pi $), we can think in terms of adding or subtracting $ 2\pi $ (which is $ 2\pi $, but in terms of $ \pi $ multiples, $ 2\pi = 2\pi $, so if we have an angle $ \alpha $, an equivalent angle can be $ \alpha - 2k\pi $ or $ \alpha + 2k\pi $ for integer $ k $.

Wait, let's check the options. The angle is $ 5.7\pi $. Let's find a coterminal angle by subtracting $ 6\pi $ (since $ 6\pi = 2\times3\pi $, but actually, the period for angles in terms of $ \pi $ when considering coterminal angles with $ \pi $ as the unit? Wait, no, the standard period for angles is $ 2\pi $ radians. So to find a coterminal angle, we can add or subtract $ 2\pi $ (or $ 4\pi $, $ 6\pi $, etc.) from the given angle.

Let's subtract $ 6\pi $ from $ 5.7\pi $: $ 5.7\pi-6\pi = - 0.3\pi $. Wait, but let's check the options. Option D is $ - 0.3\pi $. Wait, but let's verify again. Wait, maybe I made a mistake. Wait, another way: two angles are coterminal if their difference is a multiple of $ 2\pi $. Let's check the difference between $ 5.7\pi $ and each option:

- For option A: $ 5.7\pi - 2.7\pi = 3\pi $, which is not a multiple of $ 2\pi $ (since $ 3\pi/2\pi = 1.5 $, not an integer)
- For option B: $ 5.7\pi - (-1.7\pi)=7.4\pi $, $ 7.4\pi/2\pi = 3.7 $, not an integer
- For option C: $ 5.7\pi - (-5.7\pi)=11.4\pi $, $ 11.4\pi/2\pi = 5.7 $, not an integer
- For option D: $ 5.7\pi - (-0.3\pi)=6\pi $, $ 6\pi/2\pi = 3 $, which is an integer. So $ 5.7\pi $ and $ - 0.3\pi $ are coterminal (since their difference is $ 6\pi $, a multiple of $ 2\pi $). Wait, but let's check again. Wait, $ 5.7\pi + 2\pi k $, let's find $ k $ such that $ 5.7\pi + 2\pi k $ is one of the options. Let's try $ k = - 3 $: $ 5.7\pi + 2\pi(-3)=5.7\pi - 6\pi=-0.3\pi $, which is option D. Wait, but let's check the other way. Wait, maybe I made a mistake in the first approach. Let's see:

Wait, the angle $ 5.7\pi $ radians. Let's convert it to a coterminal angle by subtracting $ 2\pi $ (or $ 4\pi $, $ 6\pi $) until we get an angle in the options. Let's subtract $ 6\pi $ (since $ 6\pi = 2\times3\pi $, but $ 2\pi $ is the period, so $ 6\pi $ is $ 3 $ periods). So $ 5.7\pi - 6\pi=-0.3\pi $, which is option D. Wait, but let's check the options again. Option D is $ - 0.3\pi $, so that's the equivalent angle.

Wait, but let's check another way. The general formula for coterminal angles is $ \theta + 2k\pi $, where $ k $ is an integer. So for $ \theta = 5.7\pi $, let's find $ k $ such that $ 5.7\pi + 2k\pi $ is in the options. Let's solve for $ k $:

For option D: $ 5.7\pi + 2k\pi=-0.3\pi $

Subtract $ 5.7\pi $ from both sides: $ 2k\pi=-0.3\pi - 5.7\pi=-6\pi $

Divide both sides by $ 2\pi $: $ k = - 3 $, which is an integer. So that works. So the equivalent angle is $ - 0.3\pi $, which is option D.

Wait, but let's check the other options. Option A: $ 2.7\pi $. Let's see $ 5.7\pi - 2.7\pi = 3\pi $, which is not a multiple of $ 2\pi $ (since $ 3\pi/2\pi = 1.5 $, not integer). Option B: $ -1.7\pi $. $ 5.7\pi - (-1.7\pi)=7.4\pi $, $ 7.4\pi/2\pi = 3.7 $, not integer. Option C: $ -5.7\pi $. $ 5.7\pi - (-5.7\pi)=11.4\pi $, $ 11.4\pi/2\pi = 5.7 $, not integer. So the correct answer is D.

# Explanation:
To find an equivalent angle to $ 5.7\pi $, we use the concept of coterminal angles: two angles are coterminal if their difference is a multiple of $ 2\pi $ (i.e., $ \theta_{\text{coterminal}}=\theta + 2k\pi $, where $ k $ is an integer).

We test each option by solving $ 5.7\pi + 2k\pi=\text{option} $ for integer $ k $:

- For option D ($ -0.3\pi $):
  $ 5.7\pi + 2k\pi=-0.3\pi $
  $ 2k\pi=-6\pi $
  $ k = -3 $ (an integer), so $ 5.7\pi $ and $ -0.3\pi $ are coterminal.

# Answer:
D. $ -0.3\pi $

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Explanation:

Question 17

Step1: Recall the arc length formula \( s = r\theta \) (where \( s \) is arc length, \( r \) is radius, \( \theta \) is angle in radians)

Step2: Substitute the values into the formula

Step1: Recall the arc length formula \( s = r\theta \), solve for \( \theta \): \( \theta=\frac{s}{r} \)

Step2: Substitute the values into the formula

Step1: Recall the arc length formula \( s = r\theta \), solve for \( r \): \( r=\frac{s}{\theta} \)

Step2: Substitute the values into the formula

Answer:

Question 18