Problem solving - Which will run out first?

Question

I have a problem solving question I've been having difficulty wrapping my head around.

Let's say there are 5 different coal power stations

each uses up coal at a different rate

each has a different amount of coal reserve

each gets more coal shipped in a different rate.

Which will run out of coal first?

Answer 1

To give an algebraic solution, complementing the geometric approach, assume each coal station has $c_0$ coal in the beginning and is supplied at rate $r_i$ while using at rate $r_u$. We assume $r_u > r_i$ otherwise they never run out of coal and can be eliminated from consideration.

Then, $$ c_t = c_0 - (r_u - r_i) t, $$ and so $$c_t = 0 \implies t = \frac{c_0}{r_u - r_i}.$$

This needs to be computed for each one, and the minimum one should be selected.

Answer 2

If you plot the amount of coal at each station as a function of time you will see straight lines (with negative slopes). See which one crossed the time axis first.

You can of course do this with simple algebra but the picture may be informative.