A sector of a circle of radius 40 mi has an area of 1100 mi2. Find the central angle (in radians) of the sector.

To find the central angle \(\theta\) (in radians) of a sector, you can use the formula for the area of a sector:

\[
\text{Area of the sector} = \frac{1}{2} r^2 \theta
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.

Given:
- Area of the sector = 1100 square miles,
- Radius \( r = 40 \) miles.

We can plug these values into the formula and solve for \(\theta\):

\[
1100 = \frac{1}{2} \times 40^2 \times \theta
\]

Simplifying the equation:

\[
1100 = \frac{1}{2} \times 1600 \times \theta
\]

\[
1100 = 800 \times \theta
\]

\[
\theta = \frac{1100}{800} = \frac{11}{8} = 1.375 \text{ radians}
\]

So, the central angle \(\theta\) is \(1.375\) radians.

Would you like to know more details or have any questions? Here are some related questions:

1. What is the formula to convert an angle in radians to degrees?
2. How would the central angle change if the radius was doubled?
3. How do you calculate the length of the arc in a sector?
4. What is the relationship between the central angle and the circumference of the circle?
5. How do you find the area of a sector if the central angle is given in degrees?

**Tip:** Remember that radians are a dimensionless measure, directly relating the arc length to the radius.

Discover how to find the central angle (in radians) of a circle sector with a radius of 40 miles and an area of 1100 square miles. This problem involves using the formula for the area of a sector and understanding radians in geometry. Suitable for Grades 10-12 students.

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