Skip designs tracks for amusement park rides. For a new design, the track will be elliptical. If the ellipse is placed on a large coordinate grid with its center at (0, 0), the equation is x^2/3600+y^2/8100=1 models the path of the track. The units are given in yards. how long is the major axis of the track? Explain how you found the distance.
The equation x^2/3600+y^2/8100=1 represents an ellipse with center at (0, 0), which means that the x and y axes intersect at the center of the ellipse. The major axis of the ellipse is the longest diameter that passes through the center and has endpoints on the ellipse. The major axis is also parallel to the x-axis.
To find the length of the major axis, we need to find the distance between the two points on the x-axis where the ellipse intersects the x-axis. Setting y=0 in the equation x^2/3600+y^2/8100=1 gives x^2/3600=1, which simplifies to x=±60. Therefore, the two points of intersection are (-60, 0) and (60, 0).
The distance between these two points is the length of the major axis. Using the distance formula, we have:
d = sqrt((60 - (-60))^2 + (0 - 0)^2)
d = sqrt(120^2)
d = 120
Therefore, the major axis of the track is 120 yards long.
To find the length of the major axis of the elliptical track, we need to determine the distance between the two farthest points on the x-axis.
The equation of the ellipse is x^2/3600 + y^2/8100 = 1, where (0,0) represents the center of the ellipse.
Since the center of the ellipse is at (0,0) and the major axis lies on the x-axis, the distance between the center (0,0) and the farthest point on the x-axis will give us the length of the major axis.
The x-coordinate of the farthest point on the x-axis can be found by substituting y=0 into the equation:
x^2/3600 + 0 = 1
x^2/3600 = 1
x^2 = 3600
x = √3600
x = 60
Therefore, the length of the major axis is 2 times the distance from the center to the farthest point on the x-axis, which is 2 * 60 = 120 yards.