To find the area irrigated by the sprinkler system, we can treat the area as a sector of a circle. The area $A$ of a sector of a circle is given by:

$A = \frac{\theta}{360^\circ} \times \pi r^2$

where:

• $\theta$ is the central angle in degrees.
• $r$ is the radius of the circle (the length of the sprinkler pipe).

Given:

• $r = 400$ ft
• $\theta = 220^\circ$

Substituting these values into the formula:

$A = \frac{220^\circ}{360^\circ} \times \pi \times (400)^2$

$A = \frac{220}{360} \times \pi \times 160000$

$A = \frac{11}{18} \times \pi \times 160000$

$A \approx 0.6111 \times 502654.82 \text{ square feet}$

$A \approx 307,876 \text{ square feet}$

So, the area irrigated by the system is approximately 307,876 square feet.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the area change if the sprinkler pipe length was doubled?
2. What would be the area if the angle was reduced to 180°?
3. How does changing the angle θ affect the area irrigated?
4. If the radius was shortened to 200 ft, what would the new area be?
5. How is the formula for the area of a sector derived?

Tip: Always ensure the angle is in degrees when using this formula for a sector of a circle.