I don't see an analytical solution. For a numerical solution, consider the following:
With
\begin{align*}
m &= 100 + 200 \cdot \frac{\beta}{\beta+a} \\
50 &= m \cdot e^{-19.9a} - 200 \cdot \frac{\beta}{\beta+a} \cdot e^{19.9 \cdot \beta}
\end{align*}
Where \beta = \ln(1/2) / 26.8 \approx...
Actaully, there is sufficient information. Recall the 4.5 dimension.
OP indicates that radius is tangent at upper left point (let's call it point "A").
Let's use that point (A) as an origin in the x,y plane.
The equation of a circle is (x-a)^2 + (y-b)^2 = r^2 where point (a,b) is the center of...
You may want to look at something like
http://en.wikipedia.org/wiki/Binomial_series
Assuming neither x or y are zero (and both are positive), I would recommend factoring out the larger of x or y and let your task reduce to that of finding (1+z)^{1/2} with z<1.
For example, assume y < x...
Maybe I'm missing something, but, if the user wants Cv and Cp from inputs R and Gamma, why not just use the last two equations (3) and (4).
Equation (4) is just a re-write of equation (1). Also, Equation (3) is just a re-write of equation (2) using equation (1) (I believe equation 3 is...
eipeplusone - as Hallsofivy mentioned - expand his equation on the left hand side in terms of a polynomial in x making sure you multiply a,b, and c through. Now collect terms of a, b, and c into coefficients to this polynomial. Equate those coefficients on left to like powers of known...
Drudge - I plugged the numbers into your suggested application (from www.alcula.com) and I get the Excel results. It looks like you switched the x and y data sets. Enter the x values first, then the y values, and you should get the Excel results.
Your function "f(x)" does not agree with your formula at the top of your post. Is the last term in the bracket term of your equation "T_1 * T_2" or "T_1 + T_2".