Building a mathematical bridge angles between bank and bridge support?

firstly, your quadratic function y=-0.0113(x-33)^2+15.5 does not cross (4,6). It crosses (4,5.9967) however. it seems that this -0.0113 comes from a rounded fractional number. So before working on your problem, I will change this -0.0113 into -95/8410 so we will get precise calculation later. (-95/8410 = -0.0112960760998810939357907253269916765...
so the function will be y = -95/8410(x-33)^2 + 15.5. Check it, it will precisely cross (4,6).

your problem can be illustrated like this :
http://wolframalpha.com/input/?i=-(3%2F2)*x%2B12+%3D+-95%2F8410(x-33)^2%2B15.5
so you need to find the angle between on (4,6), between the bank (blue line) and bridge support (red line).

the easiest way to compute the degree is to transform your by substituting like this :
y = -95/8410(x-33)^2 + 15.5
y=-(2/3)x +12
substitute y in the first equation with the second one, become :
-(19 x^2)/1682+(3777 x)/1682-8.80143 = 0
then, your problem is transformed and becoming like this :
http://wolframalpha.com/input/?i=-(19+x^2)%2F1682%2B(3777+x)%2F1682-8.80143+%3D+0
now, your river bank will be the axis line, and the bridge support is the blue line one. It crosses the axis at x = 4 and x=194.789, just the same value with the problem before being transformed.

Note that this transformation is needed only to simplify our ways to find the degree between the river bank and the bridge support. The real problem is still the one before being transformed.

now, to find the angle (suppose the angle is A), we need to find the slope at x = 4. We know that slope = tangent A :)

so, compute the first derivative of the function :
http://wolframalpha.com/input/?i=d%2Fdx+-(19+x^2)%2F1682%2B(3777+x)%2F1682-8.80143
we get (3777-38 x)/1682

x = 4, we get the slope : 125/58

so, 125/58 = tangent A
A = arcustangent 125/58 = 1.13636118451391235581076226149719682936... radians = 65.11 deg
http://wolframalpha.com/input/?i=atan+125%2F58

hope this helps :)