Math / Geometry

QuestionThe radius of a circle is 2 feet. Central angle $A O B$ cuts off arc $A B$ . The length of arc $A B$ is $\frac{\pi}{6}$ yards. What is the radian measure of angle $A O B$ ? $\frac{\pi}{12}$ $\frac{\pi}{4}$ $\frac{4}{\pi}$ $\frac{12}{\pi}$

Studdy Solution

STEP 1

1. The radius of the circle is $2$ feet.
2. The length of arc $AB$ is $\frac{\pi}{6}$ yards.
3. We need to find the radian measure of angle $AOB$ .

STEP 2

1. Convert the arc length from yards to feet.
2. Use the formula for arc length to find the radian measure of angle $AOB$ .
3. Solve for the radian measure.

STEP 3

Convert the arc length from yards to feet:
Since $1$ yard is $3$ feet, the arc length in feet is:
$\frac{\pi}{6} \text{ yards} \times 3 \text{ feet/yard} = \frac{\pi}{2} \text{ feet}$

STEP 4

Use the formula for arc length, which is $L = r \theta$ , where $L$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.
Substitute the known values into the formula:
$\frac{\pi}{2} = 2 \times \theta$

STEP 5

Solve for the radian measure $\theta$ :
$\theta = \frac{\pi}{2} \div 2$ $\theta = \frac{\pi}{4}$
The radian measure of angle $AOB$ is:
$\boxed{\frac{\pi}{4}}$

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QuestionThe radius of a circle is 2 feet. Central angle $A O B$ cuts off arc $A B$ . The length of arc $A B$ is $\frac{\pi}{6}$ yards. What is the radian measure of angle $A O B$ ? $\frac{\pi}{12}$ $\frac{\pi}{4}$ $\frac{4}{\pi}$ $\frac{12}{\pi}$

Studdy Solution

STEP 1

1. The radius of the circle is $2$ feet.
2. The length of arc $AB$ is $\frac{\pi}{6}$ yards.
3. We need to find the radian measure of angle $AOB$ .

STEP 2

1. Convert the arc length from yards to feet.
2. Use the formula for arc length to find the radian measure of angle $AOB$ .
3. Solve for the radian measure.

STEP 3

Convert the arc length from yards to feet:
Since $1$ yard is $3$ feet, the arc length in feet is:
$\frac{\pi}{6} \text{ yards} \times 3 \text{ feet/yard} = \frac{\pi}{2} \text{ feet}$

STEP 4

Use the formula for arc length, which is $L = r \theta$ , where $L$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.
Substitute the known values into the formula:
$\frac{\pi}{2} = 2 \times \theta$

STEP 5

Solve for the radian measure $\theta$ :
$\theta = \frac{\pi}{2} \div 2$ $\theta = \frac{\pi}{4}$
The radian measure of angle $AOB$ is:
$\boxed{\frac{\pi}{4}}$

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QuestionThe radius of a circle is 2 feet. Central angle AOBA O BAOB cuts off arc ABA BAB. The length of arc ABA BAB is π6\frac{\pi}{6}6π​ yards. What is the radian measure of angle AOBA O BAOB ? π12\frac{\pi}{12}12π​ π4\frac{\pi}{4}4π​ 4π\frac{4}{\pi}π4​ 12π\frac{12}{\pi}π12​

Studdy Solution

STEP 1

1. The radius of the circle is 2 2 2 feet.2. The length of arc AB AB AB is π6 \frac{\pi}{6} 6π​ yards.3. We need to find the radian measure of angle AOB AOB AOB.

STEP 2

1. Convert the arc length from yards to feet.2. Use the formula for arc length to find the radian measure of angle AOB AOB AOB.3. Solve for the radian measure.

STEP 3

Convert the arc length from yards to feet:Since 1 1 1 yard is 3 3 3 feet, the arc length in feet is:π6 yards×3 feet/yard=π2 feet \frac{\pi}{6} \text{ yards} \times 3 \text{ feet/yard} = \frac{\pi}{2} \text{ feet} 6π​ yards×3 feet/yard=2π​ feet

STEP 4

Use the formula for arc length, which is L=rθ L = r \theta L=rθ, where L L L is the arc length, r r r is the radius, and θ \theta θ is the angle in radians.Substitute the known values into the formula:π2=2×θ \frac{\pi}{2} = 2 \times \theta 2π​=2×θ

STEP 5

Solve for the radian measure θ \theta θ:θ=π2÷2 \theta = \frac{\pi}{2} \div 2 θ=2π​÷2 θ=π4 \theta = \frac{\pi}{4} θ=4π​ The radian measure of angle AOB AOB AOB is:π4 \boxed{\frac{\pi}{4}} 4π​​

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QuestionThe radius of a circle is 2 feet. Central angle $A O B$ cuts off arc $A B$ . The length of arc $A B$ is $\frac{\pi}{6}$ yards. What is the radian measure of angle $A O B$ ? $\frac{\pi}{12}$ $\frac{\pi}{4}$ $\frac{4}{\pi}$ $\frac{12}{\pi}$

1. The radius of the circle is $2$ feet.
2. The length of arc $AB$ is $\frac{\pi}{6}$ yards.
3. We need to find the radian measure of angle $AOB$ .

1. Convert the arc length from yards to feet.
2. Use the formula for arc length to find the radian measure of angle $AOB$ .
3. Solve for the radian measure.

Convert the arc length from yards to feet:
Since $1$ yard is $3$ feet, the arc length in feet is:
$\frac{\pi}{6} \text{ yards} \times 3 \text{ feet/yard} = \frac{\pi}{2} \text{ feet}$

Use the formula for arc length, which is $L = r \theta$ , where $L$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.
Substitute the known values into the formula:
$\frac{\pi}{2} = 2 \times \theta$

Solve for the radian measure $\theta$ :
$\theta = \frac{\pi}{2} \div 2$ $\theta = \frac{\pi}{4}$
The radian measure of angle $AOB$ is:
$\boxed{\frac{\pi}{4}}$