QuestionCalculate the central angle $\theta$ (in degrees) for a circle with radius 15 inches and arc length 10 inches. Round to the nearest hundredth.

Studdy Solution

STEP 1

Assumptions1. The radius of the circle is15 inches. The arc length of the sector is10 inches3. We are looking for the measure of the central angle $\theta$ in degrees4. The formula to find the central angle when radius and arc length are given is $\theta = \frac{arc\, length}{radius}$ , where $\theta$ is in radians

STEP 2

First, we need to find the central angle in radians. We can do this by dividing the arc length by the radius.
$\theta = \frac{arc\, length}{radius}$

STEP 3

Now, plug in the given values for the arc length and radius to calculate the central angle in radians.
$\theta = \frac{10\, inches}{15\, inches}$

STEP 4

Calculate the central angle in radians.
$\theta = \frac{10}{15} =0.67\, radians$

STEP 5

Now that we have the central angle in radians, we need to convert this to degrees. The conversion factor is $\frac{180}{\pi}$ degrees per radian.
$\theta =0.67\, radians \times \frac{180}{\pi}\, degrees/radian$

STEP 6

Calculate the central angle in degrees.
$\theta =0.67 \times \frac{180}{\pi} =38.37\, degrees$

STEP 7

Round the central angle to the nearest hundredth of a degree.
$\theta =38.37\, degrees \approx38.37\, degrees$ The measure of the central angle $\theta$ for a sector of a circle with a radius of15 inches and arc length of10 inches is approximately38.37 degrees.

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QuestionCalculate the central angle $\theta$ (in degrees) for a circle with radius 15 inches and arc length 10 inches. Round to the nearest hundredth.

Studdy Solution

STEP 1

STEP 2

First, we need to find the central angle in radians. We can do this by dividing the arc length by the radius.
$\theta = \frac{arc\, length}{radius}$

STEP 3

Now, plug in the given values for the arc length and radius to calculate the central angle in radians.
$\theta = \frac{10\, inches}{15\, inches}$

STEP 4

Calculate the central angle in radians.
$\theta = \frac{10}{15} =0.67\, radians$

STEP 5

Now that we have the central angle in radians, we need to convert this to degrees. The conversion factor is $\frac{180}{\pi}$ degrees per radian.
$\theta =0.67\, radians \times \frac{180}{\pi}\, degrees/radian$

STEP 6

Calculate the central angle in degrees.
$\theta =0.67 \times \frac{180}{\pi} =38.37\, degrees$

STEP 7

Round the central angle to the nearest hundredth of a degree.
$\theta =38.37\, degrees \approx38.37\, degrees$ The measure of the central angle $\theta$ for a sector of a circle with a radius of15 inches and arc length of10 inches is approximately38.37 degrees.

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QuestionCalculate the central angle θ\thetaθ (in degrees) for a circle with radius 15 inches and arc length 10 inches. Round to the nearest hundredth.

Studdy Solution

STEP 1

STEP 2

First, we need to find the central angle in radians. We can do this by dividing the arc length by the radius.θ=arc lengthradius\theta = \frac{arc\, length}{radius}θ=radiusarclength​

STEP 3

Now, plug in the given values for the arc length and radius to calculate the central angle in radians.θ=10 inches15 inches\theta = \frac{10\, inches}{15\, inches}θ=15inches10inches​

STEP 4

Calculate the central angle in radians.θ=1015=0.67 radians\theta = \frac{10}{15} =0.67\, radiansθ=1510​=0.67radians

STEP 5

Now that we have the central angle in radians, we need to convert this to degrees. The conversion factor is 180π\frac{180}{\pi}π180​ degrees per radian.θ=0.67 radians×180π degrees/radian\theta =0.67\, radians \times \frac{180}{\pi}\, degrees/radianθ=0.67radians×π180​degrees/radian

STEP 6

Calculate the central angle in degrees.θ=0.67×180π=38.37 degrees\theta =0.67 \times \frac{180}{\pi} =38.37\, degreesθ=0.67×π180​=38.37degrees

STEP 7

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QuestionCalculate the central angle $\theta$ (in degrees) for a circle with radius 15 inches and arc length 10 inches. Round to the nearest hundredth.

First, we need to find the central angle in radians. We can do this by dividing the arc length by the radius.
$\theta = \frac{arc\, length}{radius}$

Now, plug in the given values for the arc length and radius to calculate the central angle in radians.
$\theta = \frac{10\, inches}{15\, inches}$

Calculate the central angle in radians.
$\theta = \frac{10}{15} =0.67\, radians$

Now that we have the central angle in radians, we need to convert this to degrees. The conversion factor is $\frac{180}{\pi}$ degrees per radian.
$\theta =0.67\, radians \times \frac{180}{\pi}\, degrees/radian$

Calculate the central angle in degrees.
$\theta =0.67 \times \frac{180}{\pi} =38.37\, degrees$