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 y? equation
2500 + 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.
To find the length of the major axis of the elliptical track, we first need to understand the equation of an ellipse in standard form:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1,
where (h,k) is the center of the ellipse, 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis.
From the given equation 2500 + 8100 = 1, we can see that this ellipse equation is not in standard form.
To convert the given equation into standard form, we have to divide both sides of the equation by 8100:
(2500/8100) + (y^2/8100) = 1,
0.3086 + (y^2/8100) = 1,
y^2 = 8100*(1 - 0.3086)
y^2 = 5541.4
Since the equation represents the y^2 value, and we are asked to find the length of the major axis, we need to find the value of 'y' for y^2 = 5541.4 :
y = sqrt(5541.4)
y ≈ 74.49
The length of the major axis of the track is equal to 2*semi-major axis = 2*y ≈ 2*74.49 ≈ 148.98 yards.
Therefore, the length of the major axis of the elliptical track is approximately 148.98 yards.