Part of the road is to be on a parabolic curve given by a function of the form y = ax^2 + bx + c where x and y are local co-ordinates.
The road alignment must pass through the following 3 points:-
x = 150m y = 190.650m
x = 300m y = 611.450m
x = 450m y = 831.141m
Substitute each pair of x,y values to produce 3 simultaneous equations with unknowns a, b and c
"Substitute each pair of x,y values to produce 3 simultaneous equations with unknowns a, b and c"
As instructed,
x = 150m y = 190.650m
a(150)^2+b(150)+c = 190.65
a(300)^2+b(300)+c = 611.450
a(450)^2+b(450)+c = 831.141
Form matrix, A=
|150^2 150 1|
|300^2 300 1|
|450^2 450 1|
X=
|a|
|b|
|c|
and
B=
|190.650|
|611.450|
|831.141|
And solve the matrix equation
AX=B.
Note: By the way, there is no automobile that can pass through this road when it has to climb 180 m in height over a distance of 150 m. (average slope of 120%). Most vehicles can only climb 20-25% slopes.