Questions: An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of 145°. What area of the field receives water? Round to two decimal places.

Solution Steps

To find the area of the field that receives water, we need to calculate the area of a sector of a circle. The formula for the area of a sector is given by:

\[ \text{Area} = \frac{\theta}{360} \times \pi \times r^2 \]

where \(\theta\) is the angle in degrees and \(r\) is the radius (distance the sprinkler sprays).

We are given the radius \( r = 35 \) feet and the angle \( \theta = 145^\circ \).

The area \( A \) of a sector of a circle is calculated using the formula:

\[ A = \frac{\theta}{360} \times \pi \times r^2 \]

Substituting the given values into the formula:

\[ A = \frac{145}{360} \times \pi \times (35)^2 \]

Calculating the area:

\[ A = \frac{145}{360} \times \pi \times 1225 \]

This results in:

\[ A \approx 1550.0705 \]

Rounding the area to two decimal places gives:

\[ A \approx 1550.07 \]

Final Answer