the towers of a bridge hung in a parabola have their  tops 30m above the  and are 200m. If the cable is 5m above the road way at the centre of the bridge. find the length of the vertical supporting cable 30m from the centre.

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$$\begin{gathered} \mathrm{Equation} \mathrm{of} \mathrm{this} \mathrm{parabola} \mathrm{will} \mathrm{be} \\ \mathrm{y}={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c} \\ \mathrm{Now} \left(0,5\right) \mathrm{is} \mathrm{on} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{the} \mathrm{parabola}, \mathrm{hence} \\ 5=0+0+\mathrm{c} \\ \Rightarrow \mathrm{c}=5 \\ \Rightarrow \mathrm{y}={\mathrm{ax}}^2+\mathrm{bx}+5 \\ \mathrm{Now} \\ \left(100,30\right) \mathrm{and} \left(-100,30\right) \mathrm{will} \mathrm{also} \mathrm{lie} \mathrm{on} \mathrm{parabola}, \mathrm{hence} \\ \mathrm{Put} \left(100,30\right) \\ 30=\mathrm{a}\left(10000\right)+\mathrm{b}\left(100\right)+5 \\ \Rightarrow 25=10000\mathrm{a}+100\mathrm{b} \\ \mathrm{Put} \left(-100,30\right) \\ 30=\mathrm{a}\left(10000\right)+\mathrm{b}\left(-100\right)+5 \\ \Rightarrow 25=10000\mathrm{a}-100\mathrm{b} \\ \mathrm{Adding} \mathrm{two} \mathrm{equations} \mathrm{we} \mathrm{get}: \\ 50=20000\mathrm{a} \\ \Rightarrow \mathrm{a}=\frac{50}{20000}=\frac{1}{400} \\ \mathrm{Subtracting} \mathrm{two} \mathrm{equations} \mathrm{we} \mathrm{get}: \\ 0=100\mathrm{b}+100\mathrm{b} \\ \Rightarrow \mathrm{b}=0 \\ \mathrm{y}={\mathrm{ax}}^2+\mathrm{bx}+5 \\ \Rightarrow \mathrm{y}=\frac{{\mathrm{x}}^2}{400}+5 \\ \mathrm{Now} \mathrm{put} \mathrm{x}=30 \mathrm{to} \mathrm{get} \mathrm{desired} \mathrm{vertical} \mathrm{length}. \\ \Rightarrow \mathrm{y}=\frac{{30}^2}{400}+5=\frac{900}{400}+5 \\ =\frac{9}{4}+5 \\ =2.25+5 \\ \mathbf{=}\mathbf{7}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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