Need help with an ellipse problem?
A bridge is built in the shape of a semielliptical arch. It has a span of 98 feet. The height of the arch 30 feet from the center is to be 11 feet. Find the height of the arch at its center.
A bridge is built in the shape of a semielliptical arch. It has a span of 98 feet. The height of the arch 30 feet from the center is to be 11 feet. Find the height of the arch at its center.
Yves From Canada
Favorite Answer
The equation of the ellipse is :
(x^2/a^2) + (y^2/b^2) = 1
We know that the span of the bridge is 98 feet. That means a = 98 / 2 = 49
The equation so far is :
(x^2/49^2) + (y^2/b^2) = 1
We know also that when x = 30 then y = 11. Just replace those values in the equation :
(30^2/49^2) + (11^2/b^2) = 1
We have to solve for b which is the height of the ellipse :
(30^2/49^2) + (11^2/b^2) = 1
(11^2/b^2) = 1 - (30^2/49^2)
b^2 = (11^2) / (1 - (30^2/49^2))
b = sqrt[(11^2) / (1 - (30^2/49^2))]
b = sqrt(193.5516)
b = 13.91 feet