A railroad tunnel is shaped like a semiellipse as shown below.
A semiellipse is shown on the coordinate plane with vertices on the x axis and one point of intersection with the positive y axis.
The height of the tunnel at the center is 27 ft and the vertical clearance must be 9 ft at a point 24 ft from the center. Find an equation for the ellipse
To find the equation for the semiellipse, we can use the standard equation of an ellipse in the form:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1
Where (h, k) represents the center coordinates of the ellipse, and a and b are the semi-major and semi-minor axes, respectively.
Given that the height of the tunnel at the center is 27 ft, this means that the semi-major axis (a) is 27 ft.
To determine the value of the semi-minor axis (b), we can use the information that the vertical clearance must be 9 ft at a point 24 ft from the center. This implies that (24, 9) lies on the ellipse.
Plugging the values into the equation, we have:
((24-h)^2)/a^2 + ((9-k)^2)/b^2 = 1
Simplifying further, we get:
((24-h)^2)/(27^2) + ((9-k)^2)/b^2 = 1
Since the ellipse intersects the positive y-axis at one point, the value of k is 0.
Thus, the equation of the semiellipse is:
((24-h)^2)/(27^2) + (y^2)/b^2 = 1
To find the equation for the semiellipse, we can use the standard form equation for an ellipse.
The standard form equation of an ellipse with a vertical major axis and center at the origin is:
(x^2 / a^2) + (y^2 / b^2) = 1
where 'a' is the semi-major axis and 'b' is the semi-minor axis.
In this case, the semiellipse has its vertices on the x-axis, so the semi-major axis is the distance from the center to one of the vertices. The vertical clearance at a point 24 ft from the center gives us the semi-minor axis.
Given that the height of the tunnel at the center is 27 ft, we know that the semi-major axis is half of the height, which is 27 / 2 = 13.5 ft.
To find the semi-minor axis, we can use the Pythagorean theorem. We have a right triangle where the height of the tunnel at a point 24 ft from the center is the height of the tunnel at the center minus the vertical clearance. So the semi-minor axis can be calculated as follows:
b^2 = (27 - 9)^2 + 24^2
b^2 = 18^2 + 24^2
b^2 = 900
b = √900
b = 30 ft
Now we have the values for the semi-major axis and semi-minor axis, so we can plug them into the standard form equation:
(x^2 / (13.5)^2) + (y^2 / (30)^2) = 1
Therefore, the equation for the given semiellipse is:
(x^2 / 182.25) + (y^2 / 900) = 1
clearly,
x^2/a^2 + y^2/27^2 = 1
at x=24, y=9, so
24^2/a^2 + 9^2/27^2 = 1
24^2/a^2 + 1/9 = 1
24^2/a^2 = 8/9
a^2 = 24^2 * 9/8 = 648
x^2/648 + y^2/729 = 1