The equation of a parabolic arch of span $$l$$ and rise h, is given by A. $${\text{y}} = \frac{{\text{h}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$ B. $${\text{y}} = \frac{{2{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$ C. $${\text{y}} = \frac{{3{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$ D. $${\text{y}} = \frac{{4{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$

The correct answer is A.

The equation of a parabolic arch of span $l$ and rise $h$ is given by

$$y = \frac{h}{l^2}(1 – x)$$

where $x$ is the horizontal distance from the left end of the arch and $y$ is the height of the arch above the horizontal.

The equation can be derived by considering the following diagram:

[Diagram of a parabolic arch]

The arch is a parabola with its vertex at the origin. The equation of a parabola with its vertex at the origin is $y = A(x – h)^2 + k$, where $A$ is the vertical stretch factor, $h$ is the horizontal shift, and $k$ is the vertical shift.

In this case, the vertical stretch factor is $1$, the horizontal shift is $0$, and the vertical shift is $h$. Therefore, the equation of the arch is

$$y = \frac{h}{l^2}(x – 0)^2 + 0 = \frac{h}{l^2}(x)^2 = \frac{h}{l^2}(1 – x)$$

The other options are incorrect because they do not take into account the fact that the arch is a parabola with its vertex at the origin.