The arch of the bridge below is half an ellipse, a "semi-ellipse". The arch is 148m long and has a height of 48m at the center. The height of the arch is 10m at point p. What is the distance from p to the center?
Draw a diagram.
Set the center of the ellipse at (0,0). Then we have semi-axes
a = 74
b = 48
x^2/74^2 + y^2/48^2 = 1
Now plug in y=10 and find x.
The distance d is then
d^2 = x^2 + 10^2
Oh, I see we're getting into some math! Well, if the arch of the bridge is a half ellipse, a "semi-ellipse," and the height of the arch is 10m at point P, we can certainly figure out the distance from P to the center.
Now, let me put on my mathematician hat and calculate this for you. 🎩
Since the height at the center is 48m and the height at point P is 10m, the difference in height between them is 48m - 10m = 38m.
We know that the center of the ellipse is the highest point, so we need to find half of that difference. Hence, the distance from point P to the center is half of 38m, which is 19 meters.
There you have it! The distance from point P to the center of the bridge is 19 meters. Keep in mind, though, that my calculations are as accurate as my jokes, so take it with a pinch of fun! 😄
To find the distance from point P to the center of the arch, we can use the properties of a semi-ellipse.
Let's denote the distance from the center to point P as d.
We know that the height of the arch is 10m at point P, and 48m at the center. Since the arch is a semi-ellipse, the height can be represented as a function of the distance from the center.
The equation of a semi-ellipse is given by:
x^2/a^2 + y^2/b^2 = 1
In this case, the height is represented by y and the distance from the center is represented by x.
We can set up a proportion using the given values:
48m / d = 10m / (148m/2)
Let's solve for d:
48m / d = 10m / 74m
Cross-multiplying:
48m * 74m = 10m * d
3566m^2 = 10m * d
Dividing by 10m on both sides:
356.6 = d
Therefore, the distance from point P to the center is approximately 356.6 meters.