This exercise involves the formula for the area of a circular sector.

A sector of a circle of radius 40 mi has an area of 1200 mi2. Find the central angle (in radians) of the sector.
 rad

To find the central angle of a sector in radians, given the area of the sector and the radius of the circle, we can use the formula for the area \(A\) of a sector:

\[
A = \frac{1}{2} r^2 \theta
\]

where:
- \(A = 1200 \, \text{mi}^2\) (area of the sector)
- \(r = 40 \, \text{mi}\) (radius of the circle)
- \(\theta\) is the central angle in radians (which we need to find).

### Step-by-Step Solution

1. Substitute the given values into the formula:

\[
1200 = \frac{1}{2} \times 40^2 \times \theta
\]

2. Simplify the equation:

\[
1200 = \frac{1}{2} \times 1600 \times \theta
\]

\[
1200 = 800 \theta
\]

3. Solve for \(\theta\) by dividing both sides by 800:

\[
\theta = \frac{1200}{800}
\]

\[
\theta = 1.5
\]

### Final Answer

The central angle of the sector is \(1.5\) radians.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the formula change if you were given the angle in degrees instead of radians?
2. What would be the area of a full circle with the same radius?
3. If the radius of the circle were doubled, how would the area of a sector change for the same central angle?
4. How can you convert this angle from radians to degrees?
5. What would be the length of the arc corresponding to this central angle?

**Tip:** When dealing with circular sectors, ensure you're using the correct formula for the unit of the angle (radians or degrees).

Learn how to calculate the central angle (in radians) of a circle sector with a radius of 40 miles and an area of 1200 square miles using the area formula of a circular sector. Detailed step-by-step solution provided.

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