Question

In a model, it is shown that an arc of a bridge is semi-elliptical with major axis horizontal. If the length of the base is $9$m and the highest part of the bridge is $3$ m from the horizontal ; the best approximation of the height of the arch, $2$m from the center of the base is

• A. $\frac{11}{4}m$
• B. $\frac{8}{3}m$
• C. $\frac{7}{2}m$
• D. $2m$

Answer 1

Answer: (B)

$\frac{8}{3}m$

Let the equation of the semi elliptical arc be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(y>0)$

Length of the major axis $=2a=9\Rightarrow a=\frac{9}{2}$

Length of the semi minor axis $=b=3$

So the equation of the arc become $\frac{4x^2}{81}+\frac{y^2}{9}=1$

If $x=2,$ then $y^2=\frac{65}{9}\Rightarrow y=\frac{1}{3}\sqrt{65}=\frac{8}{3}$ approximately.