Question

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

• A. $14$
• B. $8$
• C. $\frac{17}{2}$
• D. $\frac{23}{3}$

Answer 1

The correct option is B $8$


Bridge is in the shape of semi-ellipse.
$\therefore b=10$
and $2a=400\Rightarrow a=200$
Assume centre of the ellipse is $(0,0)$
Equation of the ellipse is
$\frac{x^2}{{200}^2}+\frac{y^2}{{10}^2}=1$

Distance of required point from centre is $200-80=120$ metres
Let depth of the point be $h,$ then $(120,h)$ lies on the ellipse.
$\Rightarrow \frac{{120}^2}{{200}^2}+\frac{h^2}{{10}^2}=1\Rightarrow h^2=64$
$\therefore h=8$ metres