A garden sprinkler sits on the ground, shooting water at a distance of 
 27 feet and rotating, thereby creating a circular watering area. Suppose the sprinkler has rotated through an angle of 4.9 radians. What area (in feet squared) of the lawn has the water covered? (Give your answer as a decimal or integer number.)

To solve the problem, we need to calculate the area of the sector of a circle that the sprinkler has watered. The formula for the area of a sector is given by:

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

Where:
- \( r \) is the radius of the circle (the distance the sprinkler shoots water),
- \( \theta \) is the angle in radians through which the sprinkler has rotated.

Given:
- \( r = 27 \) feet,
- \( \theta = 4.9 \) radians.

Plugging in these values:

\[
A = \frac{1}{2} \times 27^2 \times 4.9
\]

First, calculate \( 27^2 \):

\[
27^2 = 729
\]

Now, multiply this by \( 4.9 \):

\[
729 \times 4.9 = 3572.1
\]

Finally, multiply by \( \frac{1}{2} \):

\[
A = \frac{1}{2} \times 3572.1 = 1786.05
\]

So, the area of the lawn that has been watered is **1786.05 square feet**.

Would you like more details on any of the steps? Here are some related questions:

1. How would the area change if the sprinkler rotated through a different angle?
2. What would be the area if the radius were different, say 30 feet?
3. How would the area be affected if the angle were in degrees instead of radians?
4. What is the full area of the circular region if the sprinkler rotates a full circle?
5. How can we find the angle if the area covered and the radius are known?

**Tip:** When dealing with sectors, always ensure that the angle is in radians when using the sector area formula.

Learn how to calculate the area of a lawn covered by a sprinkler rotating 4.9 radians, with a radius of 27 feet. This problem involves applying the area of sector formula in geometry. Suitable for high school students studying geometry and radians.

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