Part of the road is to be on a parabolic curve given by a function of the form y = ax2 + 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
5.1. Substitute each pair of x,y values to produce 3 simultaneous equations with unknowns a, b and c.
5.2. Solve these simultaneous equations to determine the values of a, b and c and substitute these into the standard form to give the particular function for the road curve.
5.3. Use this function to determine the y ordinates at x = 220m and the x ordinate at y =700m
surely you can at least do part 1. ?
Try to substitute your x and y
5.1. To start, substitute each pair of x,y values into the equation y = ax^2 + bx + c, resulting in three simultaneous equations:
Equation 1: 190.650 = a(150)^2 + b(150) + c
Equation 2: 611.450 = a(300)^2 + b(300) + c
Equation 3: 831.141 = a(450)^2 + b(450) + c
5.2. Now, we need to solve the simultaneous equations to determine the values of a, b, and c.
There are multiple methods to solve these equations, such as substitution, elimination, or matrix operations. In this case, we'll use the method of matrix operations to solve the equations.
Rewriting the equations in matrix form:
| (150)^2 150 1 | | a | | 190.650 |
| (300)^2 300 1 | * | b | = | 611.450 |
| (450)^2 450 1 | | c | | 831.141 |
Let's label the matrices:
A * X = B
Using matrix operations, we can solve for X as follows:
X = A^(-1) * B
Where A^(-1) is the inverse of matrix A.
5.3. Once we have the values of a, b, and c, we can substitute them into the standard form equation y = ax^2 + bx + c to get the particular function for the road curve.
Now, we can use this function to determine the y-ordinates at x = 220m and the x-ordinate at y = 700m.
To find the y-ordinate at x = 220m, substitute x = 220m into the function and solve for y.
Similarly, to find the x-ordinate at y = 700m, rearrange the function to x = f(y), substitute y = 700m, and solve for x.
Please note that these calculations involve solving the simultaneous equations and subsequent substitutions, which may require numerical methods or calculator usage for accurate results.