Question

A railroad tunnel is shaped like a semi-ellipse. The height of the tunnel at the center is 28 ft and the total width is 50 ft. Find the vertical clearance at a point 13 to the right from the center.

Answer 1

2a=50,a=25
2b=28
b=14
Eq of ellipse is
\[\frac{ x ^{2} }{ 625 }+\frac{ y ^{2} }{196 }=1\]
when x=13
\[\frac{ 169 }{ 625 }+\frac{ y ^{2} }{ 196 }=1\]
\[\frac{ y ^{2} }{ 196 }=1-\frac{ 169 }{ 625 }=\frac{ 456 }{ 625 }\]
\[y ^{2}=\frac{ 456 }{ 625 }*196,y=\frac{ 28\sqrt{114} }{ 25 }\]
Required clearance=2y
calculate