Math / Geometry

QuestionFind the length s of the are of a circle of ractus 10 miles subterided by the centrat angle $t 9^{\circ}$ . s (arc length) $=$ $\square$ miles (Type an integer or decimal rounded to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find the length of a piece of a circle's edge, given the circle's radius and the angle that piece covers. Watch out! Remember to convert the angle from degrees to radians before calculating the arc length!

STEP 2

1. Convert the angle to radians
2. Calculate the arc length

STEP 3

Remember, $360$ degrees is equal to $2\pi$ radians.
This means $1$ degree is equal to $\frac{2\pi}{360} = \frac{\pi}{180}$ radians.
This is our conversion factor!

STEP 4

We're given an angle of $9$ degrees.
To convert it to radians, we multiply by our conversion factor: $9 \cdot \frac{\pi}{180}$ .

STEP 5

Let's simplify! $9 \cdot \frac{\pi}{180} = \frac{9\pi}{180} = \frac{\pi}{20}$ radians.
So, our angle, in radians, is $\frac{\pi}{20}$ .
Awesome!

STEP 6

The arc length $s$ is given by the formula $s = r \cdot \theta$ , where $r$ is the radius and $\theta$ is the angle in radians.
This formula is super important!

STEP 7

We know the radius $r = 10$ miles and the angle $\theta = \frac{\pi}{20}$ radians.
Let's plug these values into our formula: $s = 10 \cdot \frac{\pi}{20}$ .

STEP 8

Now, let's multiply! $s = 10 \cdot \frac{\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2}$ miles.
This is our exact arc length!

STEP 9

The problem asks for a decimal answer rounded to three decimal places.
Using the value $\pi \approx 3.14159$ , we get $s \approx \frac{3.14159}{2} \approx 1.570795$ .
Rounding to three decimal places, we get $s \approx 1.571$ miles.

STEP 10

The arc length is approximately $1.571$ miles.

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QuestionFind the length s of the are of a circle of ractus 10 miles subterided by the centrat angle $t 9^{\circ}$ . s (arc length) $=$ $\square$ miles (Type an integer or decimal rounded to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find the length of a piece of a circle's edge, given the circle's radius and the angle that piece covers. Watch out! Remember to convert the angle from degrees to radians before calculating the arc length!

STEP 2

1. Convert the angle to radians
2. Calculate the arc length

STEP 3

Remember, $360$ degrees is equal to $2\pi$ radians.
This means $1$ degree is equal to $\frac{2\pi}{360} = \frac{\pi}{180}$ radians.
This is our conversion factor!

STEP 4

We're given an angle of $9$ degrees.
To convert it to radians, we multiply by our conversion factor: $9 \cdot \frac{\pi}{180}$ .

STEP 5

Let's simplify! $9 \cdot \frac{\pi}{180} = \frac{9\pi}{180} = \frac{\pi}{20}$ radians.
So, our angle, in radians, is $\frac{\pi}{20}$ .
Awesome!

STEP 6

The arc length $s$ is given by the formula $s = r \cdot \theta$ , where $r$ is the radius and $\theta$ is the angle in radians.
This formula is super important!

STEP 7

We know the radius $r = 10$ miles and the angle $\theta = \frac{\pi}{20}$ radians.
Let's plug these values into our formula: $s = 10 \cdot \frac{\pi}{20}$ .

STEP 8

Now, let's multiply! $s = 10 \cdot \frac{\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2}$ miles.
This is our exact arc length!

STEP 9

The problem asks for a decimal answer rounded to three decimal places.
Using the value $\pi \approx 3.14159$ , we get $s \approx \frac{3.14159}{2} \approx 1.570795$ .
Rounding to three decimal places, we get $s \approx 1.571$ miles.

STEP 10

The arc length is approximately $1.571$ miles.

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QuestionFind the length s of the are of a circle of ractus 10 miles subterided by the centrat angle t9∘t 9^{\circ}t9∘. s (arc length) === □\square□ miles (Type an integer or decimal rounded to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find the length of a piece of a circle's edge, given the circle's radius and the angle that piece covers. Watch out! Remember to convert the angle from degrees to radians before calculating the arc length!

STEP 2

1. Convert the angle to radians2. Calculate the arc length

STEP 3

Remember, 360360360 degrees is equal to 2π2\pi2π radians.This means 111 degree is equal to 2π360=π180\frac{2\pi}{360} = \frac{\pi}{180}3602π​=180π​ radians.This is our **conversion factor**!

STEP 4

We're given an angle of 999 degrees.To convert it to radians, we **multiply** by our **conversion factor**: 9⋅π1809 \cdot \frac{\pi}{180}9⋅180π​.

STEP 5

Let's simplify! 9⋅π180=9π180=π209 \cdot \frac{\pi}{180} = \frac{9\pi}{180} = \frac{\pi}{20}9⋅180π​=1809π​=20π​ radians.So, our angle, in radians, is π20\frac{\pi}{20}20π​.Awesome!

STEP 6

The **arc length** sss is given by the formula s=r⋅θs = r \cdot \thetas=r⋅θ, where rrr is the **radius** and θ\thetaθ is the **angle in radians**.This formula is super important!

STEP 7

We know the **radius** r=10r = 10r=10 miles and the **angle** θ=π20\theta = \frac{\pi}{20}θ=20π​ radians.Let's plug these values into our formula: s=10⋅π20s = 10 \cdot \frac{\pi}{20}s=10⋅20π​.

STEP 8

Now, let's **multiply**! s=10⋅π20=10π20=π2s = 10 \cdot \frac{\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2}s=10⋅20π​=2010π​=2π​ miles.This is our **exact arc length**!

STEP 9

STEP 10

The arc length is approximately 1.5711.5711.571 miles.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the length s of the are of a circle of ractus 10 miles subterided by the centrat angle $t 9^{\circ}$ . s (arc length) $=$ $\square$ miles (Type an integer or decimal rounded to three decimal places as needed.)

1. Convert the angle to radians
2. Calculate the arc length

Remember, $360$ degrees is equal to $2\pi$ radians.
This means $1$ degree is equal to $\frac{2\pi}{360} = \frac{\pi}{180}$ radians.
This is our conversion factor!

We're given an angle of $9$ degrees.
To convert it to radians, we multiply by our conversion factor: $9 \cdot \frac{\pi}{180}$ .

Let's simplify! $9 \cdot \frac{\pi}{180} = \frac{9\pi}{180} = \frac{\pi}{20}$ radians.
So, our angle, in radians, is $\frac{\pi}{20}$ .
Awesome!

The arc length $s$ is given by the formula $s = r \cdot \theta$ , where $r$ is the radius and $\theta$ is the angle in radians.
This formula is super important!

We know the radius $r = 10$ miles and the angle $\theta = \frac{\pi}{20}$ radians.
Let's plug these values into our formula: $s = 10 \cdot \frac{\pi}{20}$ .

Now, let's multiply! $s = 10 \cdot \frac{\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2}$ miles.
This is our exact arc length!

The arc length is approximately $1.571$ miles.