Related Rates problem for a mobile game

Question

I'm trying to solve a problem to give me a leg up in a mobile game. (If you're curious, it's AdVenture Communist.) Here is a simplified version of what I'm working with.

A factory builds 3 machines per hour. Every machine builds 5 robots per hour. A robot can build 30 toys per hour. If you start out with 5 factories, 20 machines, and 100 robots, how long does it take to build a million toys?

I don't mind treating these as continuous variables. For instance, in ten minutes, a factory has built half a machine, but that machine is somehow already building robots (although at half capacity at that instant) and so on down the line. I'm also using Google Sheets to hold this data, so I have room for side calculations. Finally, I'm more interested in a quick calculation with 10% error than a calculation that would take hours but be down to the second.

This feels like it would be an ODE that would return a polynomial function for the number of toys in a given number of hours, but I don't see any examples quite like this in my text and I'm a little leery of trusting my own ODE instincts. (My undergraduate advisor forbade me from taking the class, as he felt it was beneath me. ^_^)

Answer 1

You might say at time $x$ hours you have $f(x)$ factories, $m(x)$ machines, $r(x)$ robots, $t(x)$ toys and make your continuity assumption. You know:

• $f(x)=5$
• $m'(x)=3f(x)=15$ and $m(0)=20$ so $m(x)=15x+20$
• $r'(x)=5m(x)=75x+100$ and $r(0)=100$ so $r(x)=\frac{75}{2}x^2+100x+100$
• $t'(x)=30r(x)=1125x^2+3000x+3000$ and $t(0)=0$ so $t(x)=375x^3+1500x^2+3000x$

That then means you would need to solve

$$1000000=375x^3+1500x^2+3000x$$

which gives $x \approx 12.48$