Okay. This is not directly related to actual space travel, but it is semi-related, and definitely a practical, actual problem I face. It's about Kerbal Space Program and SSTO spaceplanes. And I believe this is the place where I'm most likely to get an answer.
They need good thrust on ascent, requiring a chemical engine operating on Liquid Fuel + Oxidizer mix. Once in orbit, they switch to efficient, high ISp thermal nuclear engines, which run only on Liquid Fuel.
Sometimes, though, they arrive into orbit with some oxidizer to spare.
In real life scenario, at this point the crew would likely vent the excess oxidizer into space. The game engine doesn't allow this though. Either I burn the spare oxidizer in stochiometric ratio with liquid fuel, at the low ISp of the high-thrust chemical engine, or haul it along the whole journey with me, and this is the decision I must make, looking at curent tank contents (which will be different every time, so $m_{ship}, m_{lf}, m_{ox}$ are variables in the equation.
Using Tsiolkovski's Rocket Equation, this is as far as I got...
$$ \begin{align} &\Delta v = { {I_{sp} }\cdot {g_0}} ln {m_{full} \over m_{dry}} \\ &1) &\\ &m_{full} = m_{ship}+m_{ox}+m_{lf} \\ &m_{dry} = m_{ship}+m_{ox} \\ &I_{sp} = A \\ &2) \\ &m_{full_1} = m_{ship}+m_{ox}+m_{lf} \\ &m_{dry_1} = m_{ship}+m_{lf}-m_{ox}\cdot R \\ &I_{sp_1} = B \\ &m_{full_2} = m_{dry_1} = m_{ship}+m_{lf}-m_{ox} \cdot R \\ &m_{dry_2} = m_{ship} \\ &I_{sp_2} = A \\ \\ &\Delta v_1 > \Delta v_2 ?\\ \\ &R = 9/11 = 0.818\\ &A = 800 \\ &B = 305\\ &m_{lf} > m_{ox} \cdot R\\ \\ &\Delta v_1 = { {A }\cdot {g_0}} ln {{m_{ship}+m_{ox}+m_{lf}} \over {m_{ship}+m_{ox}}} = ln \left({{m_{ship}+m_{ox}+m_{lf}} \over {m_{ship}+m_{ox}}}\right)^{ {A }\cdot {g_0}}\\ \\ &\Delta v_2 = { { B }\cdot {g_0}} ln {{m_{ship}+m_{ox}+m_{lf}} \over { m_{ship}+m_{lf}-m_{ox}\cdot R}} + { {A }\cdot {g_0}} ln {{m_{ship}+m_{lf}-m_{ox}\cdot R} \over {m_{ship}}} \\ & = ln \left({{m_{ship}+m_{ox}+m_{lf}} \over { m_{ship}+m_{lf}-m_{ox}\cdot R}}\right)^{ { B }\cdot {g_0}} \cdot \left({{m_{ship}+m_{lf}-m_{ox}\cdot R} \over {m_{ship}}}\right)^{ {A }\cdot {g_0}} \\ & = ln {({m_{ship}+m_{ox}+m_{lf}})^{{B}\cdot {g_0}} \over {(m_{ship}+m_{lf}-m_{ox}\cdot R})^{{B}\cdot {g_0}}} \cdot {({m_{ship}+m_{lf}-m_{ox}\cdot R})^{{A}\cdot {g_0}} \over ({m_{ship}})^{{A}\cdot {g_0}}} \\ & = ln {({m_{ship}+m_{ox}+m_{lf}})^{{B}\cdot {g_0}} \cdot ({m_{ship}+m_{lf}-m_{ox}\cdot R})^{{(A-B)}\cdot {g_0}} \over ({m_{ship}})^{{A}\cdot {g_0}}} \\ \\ & \left({{m_{ship}+m_{ox}+m_{lf}} \over {m_{ship}+m_{ox}}}\right)^{ {A }\cdot {g_0}} > {({m_{ship}+m_{ox}+m_{lf}})^{{B}\cdot {g_0}} \cdot ({m_{ship}+m_{lf}-m_{ox}\cdot R})^{{(A-B)}\cdot {g_0}} \over ({m_{ship}})^{{A}\cdot {g_0}}}\\ & {({{m_{ship}+m_{ox}+m_{lf}} })^{ {(A-B) }\cdot {g_0}}\over ({m_{ship}+m_{ox}})^{ {A}\cdot {g_0}}} > {({m_{ship}+m_{lf}-m_{ox}\cdot R})^{{(A-B)}\cdot {g_0}} \over ({m_{ship}})^{{A}\cdot {g_0}}}\\ & {({m_{ship}})^{{A}\cdot {g_0}} \over ({m_{ship}+m_{ox}})^{ {A}\cdot {g_0}}} > {({m_{ship}+m_{lf}-m_{ox}\cdot R})^{{(A-B)}\cdot {g_0}} \over ({{m_{ship}+m_{ox}+m_{lf}} })^{ {(A-B) }\cdot {g_0}}}\\ & \left({m_{ship} \over {m_{ship}+m_{ox}}}\right)^{ {A}\cdot {g_0}} > \left({{m_{ship}+m_{lf}-m_{ox}\cdot R} \over {{m_{ship}+m_{ox}+m_{lf}} }}\right)^{ {(A-B) }\cdot {g_0}}\\ & \left({m_{ship} \over {m_{ship}+m_{ox}}}\right)^A > \left({{m_{ship}+m_{lf}-m_{ox}\cdot R} \over {{m_{ship}+m_{ox}+m_{lf}} }}\right)^ {(A-B) }\\ \end{align} $$
....and...uh, I'm swamped and probably going nowhere. Help?