Answer:
the span of the bridge is 73.7 feet
Step-by-step explanation:
The equation of an ellipse with a vertical major axis(i.e major axis parallel to y axis) is given by:
[tex]\frac{(x-h)^{2} }{b^{2} } + \frac{(y-k)^{2} }{a^{2} }= 1[/tex] a>b
where (h,k) are the coordinates of the center of the ellipse, a is the length of the major axis and b is the length of the minor axis
For this problem, the center of the ellipse (h,k) = (0,0)
Therefore:
[tex]\frac{(x)^{2} }{b^{2} } + \frac{(y)^{2} }{a^{2} }= 1[/tex]
The top of the arch is 20 feet above the ground level (the major axis), therefore a=20
length of the major axis = 2a= 2*20 = 40
[tex]\frac{x^{2} }{b^{2} } + \frac{y^{2} }{20^{2} }= 1\\\frac{x^{2} }{b^{2} } + \frac{y^{2} }{400 }= 1[/tex]
The coordinates of the ellipse (x,y) = (28,13)
[tex]\frac{28^{2} }{b^{2} } + \frac{13^{2} }{400 }= 1[/tex]
[tex]\frac{28^{2} }{b^{2} } + 0.4225= 1[/tex]
[tex]\frac{784 }{b^{2} } = 0.5775[/tex]
b² ≅ 1358
b≅36.85
Length of minor axis (2b) = 73.7 feet.
the span of the bridge is 73.7 feet
