[amp_mcq option1=”$${\text{y}} = \frac{{\text{h}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$” option2=”$${\text{y}} = \frac{{2{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$” option3=”$${\text{y}} = \frac{{3{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$” option4=”$${\text{y}} = \frac{{4{\text{h}}}}{{{l^2}}} \times \left( {1 – {\text{x}}} \right)$$” correct=”option1″]
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.