To design a circular garden that meets the client's specifications, we can follow these steps:
Step 1: Determine the coordinates of the center of the circular garden.
Given that the circular garden must be located 10 feet from one of the corners of the rectangular plot of land, we can assume that the center of the circular garden will be at a point (x, y) that is 10 feet away from one of the corners. Let's say the rectangular plot of land has corners at (0, 0), (50, 0), (50, 60), and (0, 60) (assuming a coordinate system with the bottom left corner of the plot as the origin). If the circular garden is 10 feet away from the corner at (0, 0), then the center of the circular garden will be at (x, y) = (10, 10).
Step 2: Write the equation of the circle.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values for the center of the circular garden and the diameter of 40 feet (which gives a radius of 20 feet), we get:
(x - 10)^2 + (y - 10)^2 = 20^2
Step 3: Ensure that the circular garden fits within the rectangular plot of land.
Since the circular garden must fit within the rectangular plot of land, we need to make sure that the coordinates of the center of the circular garden and the diameter of the circle do not exceed the dimensions of the rectangular plot. In this case, with a rectangular plot of 50 feet by 60 feet, the coordinates of the center of the circular garden at (10, 10) and a diameter of 40 feet (radius of 20 feet) would fit within the rectangular plot.
Step 4: Calculate the area of the circular garden.
The area of a circle with radius r is given by the formula:
Area = π * r^2
In this case, with a radius of 20 feet, the area of the circular garden would be:
Area = π * 20^2 = 400π square feet
Step 5: Ensure that the area of the circular garden meets the minimum requirement.
Given that the area of the circular garden must be at least 1256.64 square feet, we can check that the calculated area of 400π square feet meets this requirement.
In conclusion, the circular garden with a center at (10, 10) and a diameter of 40 feet, designed using the distance formula and equation of the circle, would fit within the rectangular plot of land and have an area of 400π square feet, which satisfies the minimum requirement of at least 1256.64 square feet.