Well, your original post seems to indicate you've gotten as far as forming a cubic, specifically this:
\(\displaystyle 50t^3+0t^2+16t-64000=0\)
Then, of course, you can factor out a two, leaving:
\(\displaystyle 25t^3+0t^2+8t-32000=0\)
But from here, you're left with having to factor a cubic. In general, there's really no "easy" way to do that, although there are some methods that are easier than others. A good first place to start might be to look for any rational roots, using the
Rational Roots Theorem. But that generally won't give you all three roots, unless all three happen to be rational (hint: they don't, in this case). The most straightforward way to factor a cubic is to use the
cubic formula. This will give you exact answers, but it's ugly and computationally nasty. So, probably not a good option.
A better method might be to try the
Newton-Raphson method. You posted this to the "Calculus" sub-forum, so I'm assuming you know about derivatives enough to be able to use that method (I only say this because sometimes people accidentally post things to the wrong sub-forum). Using a graphing calculator, you can see that there is a root somewhere around 10, so that would be a good x
1 to start with.
Edit: Oh, wow. It took me so long to type out this message that a whole bunch of new posts came in the meantime. It seems like you've already gotten most of the way to the solution. I'll still leave my post here for posterity's sake though.
