Say I need to produce 10 billion widgets by the end of the year. But instead of directly producing widgets, I actually produce widget-makers - each of which can produce 10 widgets/min.
What I'd love to be able to do is to calculate the minimum necessary production curve such that I could look at any given point and determine the minimum necessary production rate to likely achieve the goal by the deadline: "As of Aug 1, I am producing 50 widgets/sec and have already produced 15,000,000 widgets. [Run calculation] Ok, so this says that by Aug 30 at 17:00:00, I need to be producing 65 widgets/sec in order to still be on pace to reach 10 billion by end of year." (or whatever the numbers would be)
I've been playing around with some different approaches, but I hit dead ends. I know it's not really a standard exponential growth problem. In fact, I think it's mostly linear. But it's not simply linear as there are like 5 variables as I see it.
I've been poking at this some more since I started this post. Here's my progression of thought, starting with a (very) rough equation:
a = widgets **a**lready made
Wm = number of **W**idget-**m**akers
p = **p**roduction of widgets/sec/widget-maker (constant)
d = seconds until **d**eadline
t = **t**otal number of widgets needed by deadline (constant)
a + (Wm * p * d) = t
Now p and t are constant for any given problem:
a + (Wm * 0.17 * d) = 1x10^10
But as time progresses, how do I make sure that 'a' gets incremented by the amount of produced widgets since the last second? Let's see if instead of picturing some point in the middle as the starting point, I start at the beginning:
0 + (Wm * 0.17 * d) = 1x10^10
Then progressing one second forward:
(Wm * 0.17 * d) + (Wm * 0.17 * (d-1)) = 1x10^10
Then one more second:
(Wm * 0.17 * d) + (Wm * 0.17 * (d-1)) + (Wm * 0.17 * (d-2)) = 1x10^10
Now it's starting to look like some kind of progressing addition problem. [Is this a Riemann sum? That's literally the last thing I remember doing in math - pre-calc from 20 years ago in HS.]
Anyway, I've been trying to read up on Riemann sums, but not feeling comfortable about anything I'm trying to create from what I'm reading.
Maybe something like:
$$ \sum_{i=d}^{0} (Wm * 0.17 * i) = 1x10^{10} $$
???
I don't know. I'm far enough outside of familiar math that I don't know if I'm even in the right continent at this point. Even if this was somehow (miraculously) the right equation, I'm not sure how to go about solving it - let alone how one would go about graphing something like this. I'm willing to try to figure it out, but I really need some experts to confirm I'm on the right path or to put me on it before I spin my wheels on anything else. So I could use some help. After all, I still don't know what to do with Wm - how does that get incremented over time?
I've tried to do as much legwork myself on this as I could, but I think it's time to call in the big guns to help finish this quest. Any assistance would be appreciated.
It's not a Riemann sum, but an arithmetic progression. $$\sum_{i=0}^{d} (Wm \cdot 0.17 \cdot i) = 0.17\cdot Wm\sum_{i=1}^{d}i= \boxed{0.17\cdot Wm{d(d+1)\over 2}} $$
Well, technically, I guess it is a Riemann sum, now that I think of it, but since you're not taking the limit as the mesh of the partition goes to $0,$ it's not a useful concept here.