Question

An underpass bridge is in the shape of a semi ellipse. It is $400$ m long and has a maximum depth $10$ m at the middle point. The depth of the bridge at a point which is at a distance of $80$ m form one end is m

Answer 1

Bridge is in the shape of semi ellipse.
$\therefore b=10,2a=400\Rightarrow a=200$
Equation of the ellipse is
$\frac{x^2}{{200}^2}+\frac{y^2}{{10}^2}=1$
Centre of the ellipse is $(0,0)$
Distance of required point from centre is $200-80=120$ m
Let depth of the point is $h,$ then $(120,h)$ lies on the ellipse.
$\Rightarrow \frac{{120}^2}{{200}^2}+\frac{h^2}{{10}^2}=1\Rightarrow h^2=64\therefore h=8\text{m}$