An elliptically shaped garden is surrounded by a wood walkway. The garden is 15 meters long and 8 meters wide. The walkway is 2 meters wide. Find an equation describing the ellipse that includes both the garden and the walkway.
To find an equation describing the ellipse that includes both the garden and the walkway, we can start by considering the dimensions of the overall shape.
Let's denote the semi-major axis of the ellipse as 'a' and the semi-minor axis as 'b'. The center of the ellipse will be at the midpoint of the long side of the garden (15 meters) and the short side (8 meters).
The radius of the entire shape (garden + walkway) will be the sum of the semi-major axis 'a' and the width of the walkway (2 meters). Thus, the radius will be 15/2 + 2 = 11.5 meters.
Since the long side of the garden is aligned with the x-axis, the equation of the ellipse can be written as:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Where (h, k) represents the center of the ellipse.
Given that the radius (r) is 11.5 meters, a = 15/2 + 2 = 11.5 meters, and b = 8/2 + 2 = 6 meters.
Since the center of the ellipse is at the midpoint of the garden (15 meters) and the walkway (2 meters), the center would be at (15/2 + 2, 0) = (10.5, 0).
Substituting these values into the equation, we get:
(x - 10.5)^2 / 11.5^2 + y^2 / 6^2 = 1
Thus, the equation describing the ellipse that includes both the garden and the walkway is:
(x - 10.5)^2 / 11.5^2 + y^2 / 6^2 = 1.
To find the equation of the ellipse that includes both the garden and the walkway, we need to determine the coordinates of the center and the radii of the ellipse.
Step 1: Find the center of the ellipse
The center of the ellipse is the midpoint of the garden dimensions.
The x-coordinate of the center: (15 meters / 2) = 7.5 meters
The y-coordinate of the center: (8 meters / 2) = 4 meters
So, the center is located at (7.5, 4).
Step 2: Determine the radii of the ellipse
The major radius of the ellipse is the half of the length of the garden plus the width of the walkway:
Major radius = (15 meters / 2) + 2 meters = 7.5 meters + 2 meters = 9.5 meters
The minor radius of the ellipse is the half of the width of the garden plus the width of the walkway:
Minor radius = (8 meters / 2) + 2 meters = 4 meters + 2 meters = 6 meters
Step 3: Use the general equation of an ellipse
The equation of an ellipse with center (h, k), major radius a, and minor radius b centered at the origin (0,0) is:
((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1
Plugging in the values we found:
((x-7.5)^2 / 9.5^2) + ((y-4)^2 / 6^2) = 1
Therefore, the equation describing the ellipse that includes both the garden and the walkway is:
((x-7.5)^2 / 9.5^2) + ((y-4)^2 / 6^2) = 1
2a = 2+15+2 = 19
2b = 2+8+2 = 14
The ellipse is
x^2/a^2 + y^2/b^2 = 1
so plug in your numbers