Question $3$: $(30$ marks$)$

a$)(18$ marks$)$ A railroad horizontal curve of length $800$ meters and length of spiral is $105$ meters and had an equilibrium elevation of $120mm.$ Also given the station $125+09\mathrm{.}50m(20$ meter stations$)$ has same elevation for both rails equal to $60\mathrm{.}99m,$ if the Tangent to Spiral $($TS$)$ is at station $"x"$ and the grade from Station $125+09\mathrm{.}5$ until Spiral Tangent $($ST$)$ station is $"$Y$"($See values of $"x"$ and $"$Y$"$ for various students' groups in the table below$),$ determine

i$)(6$ marks$)$ The elevation of inner and outer rail at station $133+05\mathrm{.}50$

ii$)(6$ marks$)$ The elevation of inner and outer rail at station $148+19\mathrm{.}51$

iii$)(6$ marks$)$ The elevation of inner and outer rail at station $178+10\mathrm{.}00$

$\backslash$table$[[\backslash$table$[[$Students fifth and seventh numbers$],[$from the left $($example of number$],[$in red and bold: $1161840$ the$],[$number is $80$ in this example$)]],\backslash$table$[[$Tangent to Spiral $($TS$)],[$Station $"$X$"]],\backslash$table$[[$The grade of the$],[$entire rail section$],[($from station$],[125+09\mathrm{.}50$ to Spiral$],[$Tangent $($ST$)"$Y$"]]],[A=$ Between $00$ and $24($inclusive$),130+11\mathrm{.}50,-0\mathrm{.}50\%$