Question

An arc is in the form of a semi-ellipse. It is $8m$ wide and $2m$ high at the centre. Find the approximate height of the arc at a point $1.5m$ from one end.

• A. $0.56m$
• B. $1.56m$
• C. $2.30m$
• D. $4.52m$

Answer 1

Answer: (B)

The semi-elliptical arc has been shown in the figure. We choose the coordinate axes as shown in the figure.

Here $OA=\frac{1}{2}\cdot$ $8=4$ and $OB=2$
$\therefore$ The equation of the ellipse is
$\frac{x^2}{4^2}+\frac{y^2}{2^2}=1$ i.e., $x^2+4y^2=16\dots \left(i\right)$
Let $h$ metres be the height of the arc at a distance of $1.5m$ from one end, then $OM=4-1.5=2.5$, so that the point $P$ is $\left(2.5,h\right)$.
Since $P$ lies on ellipse $\left(i\right)$
$\therefore {\left(\frac{5}{2}\right)}^2+4h^2=16$
$\Rightarrow 4h^2=16-\frac{25}{4}$
$\Rightarrow h=\frac{\sqrt{39}}{4}=1.56m$ (approx.) $\left[\because h>0\right]$