A garden sprinkler sits on the ground, shooting water at a distance of
35 feet and rotating, thereby creating a circular watering area. Suppose the sprinkler has rotated through an angle of 1.6 radians. What area (in feet squared) of the lawn has the water covered? (Give your answer as a decimal or integer number.)

To determine the area of the lawn covered by the sprinkler, we use the formula for the area of a sector:

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

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

Given:
- \( r = 35 \) feet,
- \( \theta = 1.6 \) radians.

Substitute these values into the formula:

\[
A = \frac{1}{2} \times 35^2 \times 1.6
\]

First, calculate \( 35^2 \):

\[
35^2 = 1225
\]

Now, multiply this by \( 1.6 \):

\[
1225 \times 1.6 = 1960
\]

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

\[
A = \frac{1}{2} \times 1960 = 980
\]

Thus, the area of the lawn that has been watered is **980 square feet**.

Would you like further details or have any questions? Here are some related questions:

1. What would be the area if the sprinkler rotated through a full circle (2π radians)?
2. How does the angle affect the area covered, if all other factors remain constant?
3. What happens to the area if the radius is increased to 40 feet with the same angle?
4. How can we calculate the angle if the area covered and radius are known?
5. What is the perimeter of the sector covered by the sprinkler?

**Tip:** Remember, the area of a sector increases linearly with the angle in radians, so doubling the angle doubles the area covered.

Learn how to calculate the area covered by a garden sprinkler shooting water at 35 feet and rotating through 1.6 radians. Use the area of a sector formula to find the lawn area covered. Suitable for high school students studying geometry and trigonometry.

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