One method of irrigating crops is called the center picot system. This system rotates a sprinkler pipe from the center of the field to be irrigated. Suppose a farmer places one of these units at the center of a square plot of land 2500 feet on each side. With the center of this plot at the origin, the irrigator sends out water far enough to reach a point located at (4785, 1140)

A) Find an equation representing the farthest points the water can reach.

b) Find the area of the land that receives water directly

c) About what percent of the farmer's plot does not receive water directly?

well, r^2 = 4785^2+1140^2, so the circle's radius is 4919 feet.

x^2+y^2 = 4949^2

Now just use your knowledge of the areas of circles and squares to answer the questions.

A) To find an equation representing the farthest points the water can reach, we can use the equation of the circle. The center of the circle is at (0, 0) and the radius is the distance from the center to the point (4785, 1140). Using the distance formula, we can calculate the radius:

Radius = √[(4785-0)^2 + (1140-0)^2]
Radius = √[(4785^2) + (1140^2)]
Radius ≈ 5031.43

Therefore, the equation representing the farthest points the water can reach is:

x^2 + y^2 = 5031.43^2

B) To find the area of the land that receives water directly, we need to calculate the area of the sector formed by the angle subtended by the point (4785, 1140) at the center of the circle.

First, let's find the angle θ using trigonometry:

θ = arctan(1140/4785)
θ ≈ 13.36 degrees

Since the radius of the circle is 5031.43, the area of the sector is:

Area of the sector = (13.36/360) * π * (5031.43^2)

C) To figure out the percentage of the farmer's plot that does not receive water directly, we need to calculate the area of the square plot and subtract the area of the land that receives water directly. Then, we can calculate the percentage:

Area of the square plot = (2500)^2
Percentage of land not receiving water directly = [(Area of the square plot - Area of watered land)/Area of the square plot] * 100

Using the values calculated in parts A and B, we can substitute them into the formula above to find the percentage.

a) To find an equation representing the farthest points the water can reach, we need to understand the behavior of the sprinkler pipe in the center pivot system.

In a center pivot system, the sprinkler pipe rotates around the center in a circular motion, creating a circular region of irrigated land. The distance from the center to the edge of this circular region is the radius of the circle.

Given that the farm plot is a square with sides of 2500 feet, the maximum distance the water can reach is the distance from the center to any corner of the square. Using the distance formula, we can calculate the distance.

The distance from the center (0,0) to any point (x,y) on the square is given by the formula:

d = √((x-0)^2 + (y-0)^2)

For the farthest point on the square, (x,y) = (4785,1140), we have:

d = √((4785-0)^2 + (1140-0)^2)
d = √(4785^2 + 1140^2)
d = √(22867225 + 1299600)
d = √(24166825)
d ≈ 4915.8

So, the equation representing the farthest points the water can reach is:

(x-0)^2 + (y-0)^2 = 4915.8^2
x^2 + y^2 = 24166825

b) To find the area of the land that directly receives water, we need to find the area of the circular region created by the center pivot system.

The area of a circle is given by the formula: A = πr^2

Since the maximum radius (from the center to any corner of the square) is approximately 4915.8 feet, the area can be calculated as:

A = π * 4915.8^2
A ≈ π * 24166825
A ≈ 75828318.3 square feet

c) To find the percentage of the farmer's plot that does not receive water directly, we need to compare the area of the square plot with the area of the circular region of irrigated land.

The area of the square plot is given by the formula: A = side^2 = (2500)^2 = 6250000 square feet

The percentage of the plot that does not receive water directly can be calculated by subtracting the area of the circular region from the area of the square plot, dividing by the area of the square, and multiplying by 100:

Percentage = ((Area of square - Area of circle) / Area of square) * 100
Percentage = ((6250000 - 75828318.3) / 6250000) * 100
Percentage ≈ -1113.25%

Note: The negative value indicates that the area of the circular region is larger than the area of the square plot, which is not possible. Therefore, there might be an error in the calculations or the given information.