Question

An arch for a bridge over a highway is in the form of a semi ellipse. The top of the arch is 35 feet above ground​ (the major​ axis). What should the span of the bridge be​ (the length of its minor​ axis) if the height 27 feet from the center is to be 15 feet above​ ground? Round to two decimal places

Answer 1

Answer: The span of the bridge is 59.77ft

Step-by-step explanation:

The standard form of equation for an ellipse with vertical major axis is:

(x-h)^2/b^2+(y-k)^2/a^2=1,a>b, (h,k)=(x,y) coordinates of center.

For given problem:

Place center of ellipse, (0,0) at center of bridge at ground level.

Given length of vertical major axis=70=2a

a=35

a^2=1225

Equation of an ellipse with horizontal major axis with center at (0,0)

Equation of ellipse:

x^2/b^2+y^2=1

plug in coordinates of given point on ellipse(27, 15)

27^2/b^2+15^2/a^2=1

729/b^2+225/1225=1

729/b^2=1-225/1225= 0.816

b^2=729/0.816≈ 893

b≈29.88

length of minor axis=2b≈59.77 ft

span of bridge≈59.77 ft