A cable of a suspension bridge is in the form of a parabola whose span is 40 m. The roadway is 5 m below the lowest point of the cable. If an extra support is provided across the cable 30 m above the ground level, find the length of the support if the height of the pillars are 55 m.
Take the lowest point on the suspension bridge as the vertex and let it be at the origin.
$\therefore$ $\;$ Vertex $= V = \left(0,0\right)$
Let $\;$ $RR'$ $\;$ be the roadway.
$PQ$ $\;$ is the span of the bridge $= 40$ m
Let extra support be provided at the points $AA'$.
$BA =$ height of point $A$ from the roadway $= 30$ m
$B'A' =$ height of point $A'$ from the roadway $= 30$ m
$PP'$, $\;$ $QQ' =$ pillars of height $55$ m
$\because$ $\;$ $PQ = 40$ m $\implies$ $VD' = 20$ m
$\therefore$ $\;$ Coordinates of $Q = \left(20, 50\right)$
Equation of the parabola is $\;\;$ $x^2 = 4ay$ $\;\;\; \cdots \; (1)$
$\because$ $\;$ Point $Q$ lies on equation $(1)$, we have
$\left(20\right)^2 = 4 \times a \times 50$ $\implies$ $a = 2$
Substituting the value of $a$ in equation $(1)$ gives the equation of the parabola as
$x^2 = 8y$ $\;\;\; \cdots \; (2)$
Let $VC' = \ell$ m.
From the figure, $\;$ $A'C' = 25$ m
Then the coordinates of point $A'$ are $\left(\ell, 25\right)$
$\because$ $\;$ Point $A'$ lies on the parabola given by equation $(2)$, we have
$\ell^2 = 8 \times 25 = 200$ $\implies$ $\ell = 10 \sqrt{2}$
$\therefore$ $\;$ From the figure, $AA' = 2 \ell = 20 \sqrt{2}$
$\therefore$ $\;$ Length of support $= 20 \sqrt{2}$ m