I have no clue how to answer this question.
A farmer has a square plot of land. An irrigation system can be installed with the option of one large circular sprinkler, or nine small sprinklers. The farmer wants to know which plan will provide water to the greatest percentage of land in the field, regardless of the cost and the watering pattern. What advice would you give?
Let's show it by using actual numbers.
Suppose the field is 300m by 300m
area of field is 90000 m^2
area of largest circle = pi(150^2) = 22500pi = 70686 m^2
so area left not sprinkled = 19314 m^2
If we arrange the 9 smaller sprinklers in a pattern like a game of X's and O's
then each radius is 50
and each circle has an area of pi(50^2) = 7853.98
so 9 of those would give us 70686
THE SAME, WOW
The question remains, "Can the 9 circles be arranged so that their radius would be increased" ?
To determine which irrigation plan will provide water to the greatest percentage of land in the field, we can compare the area covered by each sprinkler system.
1. Large Circular Sprinkler:
The large circular sprinkler covers a circular area. To find the area of this circle, we need to determine the radius.
Since the plot of land is square, the length of one side of the square is equal to the diagonal of the square. Let's assume this side length is 's'.
Using Pythagoras theorem, we find that the diagonal 'd' of a square is √(s^2 + s^2).
Therefore, the radius 'r' of the circular area covered by the large sprinkler is equal to half of the diagonal, r = d/2 = (√(s^2 + s^2))/2 = (√2s^2)/2 = s√2/2.
The area covered by the large sprinkler is given by the formula for the area of a circle: A = πr^2 = π(s√2/2)^2 = πs^2/2.
2. Nine Small Sprinklers:
The nine small sprinklers can be arranged in a 3x3 grid, covering the entire square plot of land. Each small sprinkler will cover the same area.
The area covered by each small sprinkler is the same as the area of the circular coverage of the large sprinkler, which is πs^2/2.
Therefore, the total area covered by the nine small sprinklers is 9 * (πs^2/2) = (9πs^2)/2.
To determine which plan provides water to the greatest percentage of land, we can compare the ratio of the area covered by each plan to the total area of the square plot.
The percentage of land covered by the large circular sprinkler is (πs^2/2) / (s^2) = π/2.
The percentage of land covered by the nine small sprinklers is [(9πs^2)/2] / (s^2) = (9π)/2.
Comparing the percentages, we find that the percentage of land covered by the nine small sprinklers is greater than the percentage covered by the large circular sprinkler.
Therefore, my advice would be to choose the plan with the nine small sprinklers, as it will provide water to a greater percentage of the land in the field.
To determine which plan will provide water to the greatest percentage of land, we need to compare the coverage areas of the two irrigation plans: one large circular sprinkler and nine small sprinklers.
To calculate the coverage area of a circular sprinkler, you need to know the radius of the sprinkler head. The formula to calculate the area of a circle is A = π * r^2, where A is the area and r is the radius of the circle.
For the large circular sprinkler, you need to find the radius that covers the same area as the nine small sprinklers combined. Since you haven't provided any information about the size or dimensions of the land, we'll assume the area of the square plot to be 'A'.
Now, to calculate the radius of the large circular sprinkler, we need to calculate the area covered by the nine small sprinklers. Since each small sprinkler covers the same area, we can divide the area 'A' by 9 to find the area covered by one small sprinkler.
Once we have the area covered by one small sprinkler, we can plug it into the formula A = π * r^2 and solve for the radius (r). You can use any circle area calculator or a math software/tool to find the radius of the small sprinklers' coverage.
Once you have the radius of the small sprinklers' coverage, double-check that it's actually smaller than the side length of the square plot of land. If it's not, then multiple small sprinklers won't fit inside the square plot.
With the radius of the small sprinklers' coverage and the side length of the square plot, you can calculate the area covered by the large circular sprinkler using the formula A = π * r^2.
Now, compare the percentage of land covered by each plan. Divide the area covered by the circular sprinkler (π * r^2) by the total area of the square plot (A) and multiply by 100 to get the percentage coverage. Repeat the same for the small sprinklers and compare the percentages obtained.
Finally, advise the farmer based on which plan provides the greater percentage of coverage.
Note: This explanation assumes that the sprinklers are evenly spaced and placed inside the square plot. The watering pattern and the positioning of the sprinklers could influence the coverage, but since you mentioned it doesn't matter for this comparison, we can ignore the specific pattern for now.