An Arc On A Circle Measures $250^{\circ}$. Within Which Range Is The Radian Measure Of The Central Angle?A. $\pi$ To $\frac{3 \pi}{2}$ Radians B. $\frac{3 \pi}{2}$ To $2 \pi$ Radians C.

Understanding the Relationship Between Degrees and Radians

To determine the range within which the radian measure of the central angle falls, we need to understand the relationship between degrees and radians. The radian measure of a central angle is calculated by dividing the measure of the angle in degrees by $180^{\circ}$ and then multiplying by $\pi$ radians.

Formula for Radian Measure

The formula to convert degrees to radians is given by:

$$\text{radians} = \frac{\text{degrees}}{180^{\circ}} \times \pi$$

Applying the Formula to the Given Angle

Given that the arc on a circle measures $250^{\circ}$, we can use the formula to find the radian measure of the central angle.

$$\text{radians} = \frac{250^{\circ}}{180^{\circ}} \times \pi$$

Simplifying the Expression

To simplify the expression, we can divide the numerator and denominator by their greatest common divisor, which is $10^{\circ}$.

$$\text{radians} = \frac{25^{\circ}}{18^{\circ}} \times \pi$$

Evaluating the Expression

Now, we can evaluate the expression to find the radian measure of the central angle.

$$\text{radians} = \frac{25}{18} \times \pi$$

Approximating the Value

To approximate the value, we can multiply the fraction by $\pi$.

$$\text{radians} \approx \frac{25}{18} \times 3.14159$$

Calculating the Value

Now, we can calculate the value to find the radian measure of the central angle.

$$\text{radians} \approx 4.38722$$

Determining the Range

Since the radian measure of the central angle is approximately $4.38722$, we need to determine the range within which this value falls.

Range of Radian Measures

The range of radian measures for central angles is typically given as:

• $\pi$ radians to $\frac{3 \pi}{2}$ radians
• $\frac{3 \pi}{2}$ radians to $2 \pi$ radians

Comparing the Value to the Ranges

Now, we can compare the value of the radian measure of the central angle to the two ranges.

Conclusion

Based on the comparison, we can conclude that the radian measure of the central angle falls within the range of $\frac{3 \pi}{2}$ to $2 \pi$ radians.

Final Answer

The final answer is B. $\frac{3 \pi}{2}$ to $2 \pi$ radians.

Understanding the Relationship Between Degrees and Radians

To determine the range within which the radian measure of the central angle falls, we need to understand the relationship between degrees and radians. The radian measure of a central angle is calculated by dividing the measure of the angle in degrees by $180^{\circ}$ and then multiplying by $\pi$ radians.

Formula for Radian Measure

The formula to convert degrees to radians is given by:

$$\text{radians} = \frac{\text{degrees}}{180^{\circ}} \times \pi$$

Applying the Formula to the Given Angle

Given that the arc on a circle measures $250^{\circ}$, we can use the formula to find the radian measure of the central angle.

$$\text{radians} = \frac{250^{\circ}}{180^{\circ}} \times \pi$$

Simplifying the Expression

To simplify the expression, we can divide the numerator and denominator by their greatest common divisor, which is $10^{\circ}$.

$$\text{radians} = \frac{25^{\circ}}{18^{\circ}} \times \pi$$

Evaluating the Expression

Now, we can evaluate the expression to find the radian measure of the central angle.

$$\text{radians} = \frac{25}{18} \times \pi$$

Approximating the Value

To approximate the value, we can multiply the fraction by $\pi$.

$$\text{radians} \approx \frac{25}{18} \times 3.14159$$

Calculating the Value

Now, we can calculate the value to find the radian measure of the central angle.

$$\text{radians} \approx 4.38722$$

Determining the Range

Since the radian measure of the central angle is approximately $4.38722$, we need to determine the range within which this value falls.

Range of Radian Measures

The range of radian measures for central angles is typically given as:

• $\pi$ radians to $\frac{3 \pi}{2}$ radians
• $\frac{3 \pi}{2}$ radians to $2 \pi$ radians

Comparing the Value to the Ranges

Now, we can compare the value of the radian measure of the central angle to the two ranges.

Conclusion

Based on the comparison, we can conclude that the radian measure of the central angle falls within the range of $\frac{3 \pi}{2}$ to $2 \pi$ radians.

Final Answer

The final answer is B. $\frac{3 \pi}{2}$ to $2 \pi$ radians.

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Q: What is the relationship between degrees and radians?

A: The radian measure of a central angle is calculated by dividing the measure of the angle in degrees by $180^{\circ}$ and then multiplying by $\pi$ radians.

Q: How do I convert degrees to radians?

A: You can use the formula: $\text{radians} = \frac{\text{degrees}}{180^{\circ}} \times \pi$.

Q: What is the formula for converting degrees to radians?

A: The formula is: $\text{radians} = \frac{\text{degrees}}{180^{\circ}} \times \pi$.

Q: How do I simplify the expression when converting degrees to radians?

A: You can divide the numerator and denominator by their greatest common divisor.

Q: What is the range of radian measures for central angles?

A: The range of radian measures for central angles is typically given as: $\pi$ radians to $\frac{3 \pi}{2}$ radians and $\frac{3 \pi}{2}$ radians to $2 \pi$ radians.

Q: How do I determine the range within which the radian measure of the central angle falls?

A: You can compare the value of the radian measure of the central angle to the two ranges.

Q: What is the final answer for the given problem?

A: The final answer is B. $\frac{3 \pi}{2}$ to $2 \pi$ radians.

Q: How do I calculate the value of the radian measure of the central angle?

A: You can use the formula: $\text{radians} = \frac{\text{degrees}}{180^{\circ}} \times \pi$ and then approximate the value by multiplying the fraction by $\pi$.

Q: What is the approximate value of the radian measure of the central angle?

A: The approximate value is $4.38722$.

Q: How do I compare the value of the radian measure of the central angle to the two ranges?

A: You can compare the value to the two ranges: $\pi$ radians to $\frac{3 \pi}{2}$ radians and $\frac{3 \pi}{2}$ radians to $2 \pi$ radians.

Q: What is the conclusion for the given problem?

A: The radian measure of the central angle falls within the range of $\frac{3 \pi}{2}$ to $2 \pi$ radians.