Question

An arch is in the shape of a parabola whose axis is vertically downward and measures $80$ meters across the bottom on the ground. Its highest point is $24$ meters. Then the measure of the horizontal beam across the section at a height of $18$ meters is

• A. $55$
• B. $50$
• C. $45$
• D. $40$

Answer 1

Answer: (D)

$40$

We can write the equation of the parabola as $y=-a{x}^2+c$
The vertex of the parabola is $(0,c)$
Since, the highest point is $24$ metres.
$c=24$.
The span is $80$ metres. Hence, at $x=40$, $y=0$.
Hence,
$-1600a+24=0$
$\Rightarrow a=\frac{24}{1600}=\frac{3}{200}$
When height is $18$ metres,
$y=18$
$\Rightarrow -\frac{3}{200}{x}^2+24=18$
$\Rightarrow x^2=400$
$x=\pm 20$.
Hence, the length of the beam $=40$