Question

Open in App

Verified by Toppr

The origin of the coordinate plane is taken as the centre of the ellipse while the major axis is taken along the $x$-axis. Hence the semi-ellipse can be diagrammatically represented as,

The equation of the semi-ellipse will be of the form $a_{2}x_{2} +b_{2}y_{2} =1,y≥0$ where $a$ is the semi-major axis.

Accordingly, $2a=8$

$a=4,b=2$

Therefore, the equation of the semi-ellipse is $16x_{2} +4y_{2} =1,y≥0$ ...(1)

Let $B$ be a point on the major axis such that $AB=1.5$ m

Draw $BC$ $⊥$ $OA$

$OB=(4−1.5)$ $m$ $=2.5$ $m$

The $x$-coordinate of point $C$ is $−2.5$ $m$.

On substituting the value of $x$ with $−2.5$ in equation (1), we obtain

$16(−2.5)_{2} +4y_{2} =1$

$⇒166.25 +4y_{2} =1$

$⇒y_{2}=4(1−166.25 )$

$⇒y_{2}=4(169.75 )$

$⇒y_{2}=2.4375$

$⇒y=1.56$ (approx) $(∵y≥0)$

$∴AC=1.56$ $m$

Thus the height of the arch at a point $1.5$ m from one end is approximately $1.56$ $m$.

0

0