A falling gradient of $4\%$ meets a rising gradient of $5\%$ in a sag vertical curve. At the start of the vertical curve at point $P,$ the reduced level is $334\mathrm{.}220m$ at chainage $1350m$ as shown in Figure $2.$ At chainage $1450m$ the underside of a bridge has a reduced level of $345\mathrm{.}345m.$ The two grades are to be joined by a vertical curve giving $14m$ clearance under the bridge.

i$.$ Determine the length of the vertical curve $(L)$ and the chainage of intersection point $Q$ and tangent point $R.$

$(11$ marks$)$

ii$.$ Determine the reduced levels of points $Q$ and $R.$

$(3$ marks$)$

iii. Determine the lowest point on the vertical curve.

$(3$ marks$)$

iv$.$ By using the given table in APPENDIX B$,$ determine the reduced levels at $50$ m intervals along the curve.