Question

State the following statement is True or False

A parabolic arch has a height $18$ meters and span $24$ meters. Then the height of the arch at $8$ meters from the centre of the span is equal to 10

• A. True
• B. False

Answer 1

The correct option is A True

The parabolic arc $OACB$ is shown in the figure. $C$ is the center of the span. By taking the equation of parabola in standard form $y^2=4ax$, we get coordinates of $A$ as $(18,12)$
Hence, ${12}^2=4a\times 18$
$\Rightarrow 4a=\mathrm{8.....}\left(1\right)$
Let the height of the arc (distance from center) for a point which is at a distance of $8$ meters from the span (y-coordinate in the figure) be $h$.
Then, the coordinates of one point on the parabola becomes $(18-h,8)$
Hence, $8^2=4a\times (18-h)$
$\Rightarrow 18-h=8$
$\Rightarrow h=10$