A railroad tunnel is shaped like semi-ellipse. The height of the tunnel at the center is 69 ft and the vertical clearance must be 23 ft at a point 16ft from the center. Find an equation for the ellipse.
b = 69, a is unknown, and the point (16,23) is on it
let the equation be
x^2/a^2 + y^2/69^2 = 1
16^2/a^2 + 23^2/69^2 = 1
256/a^2 = 1 - 529/4761
256/a^2 = 8/9
8a^2 = 2304
a^2 = 288
x^2/288 + y^2/4761 = 1
Well, this is quite a tunnel conundrum we have here! Let's see if we can solve it with a touch of humor, shall we?
So, we're dealing with a semi-ellipse that's taking the shape of a railroad tunnel. Now, I'm not sure if this tunnel has a sense of humor, but let's give it a shot.
First, let's start with the center of the tunnel, which we'll call point O. We know that the height at the center, point O, is 69 ft. That's pretty high, but don't worry, we won't get tunnel vision just yet.
Now, let's move 16 ft away from the center to point A. Here, the vertical clearance is 23 ft. It seems like the tunnel is getting a bit shy and trying to maintain some personal space.
To find an equation for the semi-ellipse, we need to use a touch of math and humor. Are you ready? Here comes the punchline equation:
(x^2/16^2) + (y^2/((69^2 - 23^2)/2)^2) = 1
Well, there you have it! The equation for the semi-ellipse shaped railroad tunnel, with a height of 69 ft at the center and a vertical clearance of 23 ft at a point 16 ft from the center. Remember, even tunnels can have a sense of humor, especially when it comes to equations.
To find the equation of the ellipse, let's assume the semi-ellipse is aligned vertically with its axis of symmetry along the y-axis.
Let (x, y) be any point on the semi-ellipse.
Since the height of the tunnel at the center is 69 ft, the coordinates of the center of the semi-ellipse are (0, 69).
Let's first determine the coordinates of the point on the semi-ellipse where the vertical clearance is 23 ft.
The vertical clearance of 23 ft is measured from the center of the semi-ellipse, and the point is located 16 ft away from the center. This means that the y-coordinate of this point is 69 - 23 = 46 ft.
So, the coordinates of this point are (16, 46).
Now, we will use the equation of an ellipse with vertical major axis to find the equation of the semi-ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
where (h, k) are the coordinates of the center, a is the semi-major axis, and b is the semi-minor axis of the ellipse.
Since the semi-ellipse is aligned vertically, the center coordinates are (0, 69).
Let's find the values of a and b.
The semi-major axis a is the distance from the center to the point where the vertical clearance is 23 ft. This is equal to 16 ft.
The semi-minor axis b is the distance from the center to the highest point (69 ft) or the lowest point (46 ft) on the semi-ellipse. This is equal to (69 - 46) / 2 = 11.5 ft.
Therefore, the equation of the ellipse is:
(x - 0)^2/16^2 + (y - 69)^2/11.5^2 = 1
Simplifying, the equation becomes:
x^2/256 + (y - 69)^2/132.25 = 1
To find the equation for the ellipse, we need to consider the properties of an ellipse and use the given information.
An ellipse can be defined by its center coordinates (h, k), the length of its major axis (2a), and the length of its minor axis (2b). The standard equation for an ellipse with its center at the origin is:
(x^2/a^2) + (y^2/b^2) = 1
In this case, since the tunnel is symmetrical, we can assume the center of the ellipse is at the origin. So, h = 0, k = 0.
We are given that the height of the tunnel at the center is 69 ft, which means the major axis of the ellipse is twice this height, so 2a = 2 * 69 = 138 ft. Therefore, a = 69 ft.
We are also given that the vertical clearance must be 23 ft at a point 16 ft from the center. This point lies on the minor axis of the ellipse. So, 2 * b = 23 ft. Therefore, b = 23/2 = 11.5 ft.
Now, we have the values of a and b, so we can write the equation of the ellipse as:
(x^2/69^2) + (y^2/11.5^2) = 1
Simplifying the equation, we have:
x^2 / 4761 + y^2 / 132.25 = 1
Therefore, the equation for the ellipse shaped railroad tunnel is:
x^2 / 4761 + y^2 / 132.25 = 1