Final answer:
By applying the equation of an ellipse, the height of a semi-elliptical bridge, which is 400 meters long with a maximum height of 10 meters, at a point 80 meters from one end, is calculated to be 8 meters.
Step-by-step explanation:
A bridge is in the shape of a semi-ellipse, which dictates that the relationship between its width and height at any given point follows the equation of an ellipse. Since the bridge is 400 meters long, it can be assumed that its semi-major axis (a) is 200 meters (half the length), and the semi-minor axis (b) is the maximum height of 10 meters.
Using the information given, the height of the bridge at a point 80 meters from one end can be found by the ellipse equation (x/a)^2 + (y/b)^2 = 1, where 'x' is the distance from the center of the ellipse.
To find the height at 80 meters from one end, we first determine the distance from the center of the bridge by calculating 200 meters (semi-major axis) minus 80 meters, which equals 120 meters. Substituting 'a' with 200 meters, 'b' with 10 meters, and 'x' with 120 meters into the ellipse equation, we get:
(120/200)^2 + (y/10)^2 = 1
Solving for 'y', we determine the height 'y' at the given point:
((120^2)/(200^2)) + (y^2)/(10^2) = 1 →
(14400/40000) + (y^2)/(100) = 1 →
0.36 + (y^2)/(100) = 1 →
(y^2)/(100) = 1 - 0.36 →
y^2 = 64 →
y = 8 meters
Thus, the height of the bridge at a point 80 meters from one end is 8 meters, which corresponds to option C.