So I was wondering how long it took my iPad to go from 60% to $100\%$ since I got a new USB port. I found it took $216$ minutes, or about $3.6$ hours. My goal was to figure out the time it takes to charge $1\%$ (namely in minutes), assuming of course, the rate of charge is consistent. I also tried backtracking the math to figure out how long it should take to go from $0\%$ to $100\%$ (based on the fact that $\frac{.40}{216}=1\% \text{ per} \ 540 \ \text{minutes}$), for example, but I realized I can't trust those results if I can't figure out a conventional rate per minute!

So I figured $$\frac{0.40}{216}\propto \frac{0.01}{9}$$

I decided to represent this algebraically where the percentage of charge, $P$, is a function of time in minutes, $t$

Which should mean


So I tested this function with the actual data from above $$P(216)=\frac{0.01}{9}(216) \implies P=0.24$$

Clearly, $24\%\ne40\%$

I figured no calculus would be needed because I'm assuming rate of charge is consistent/linear. What am I missing here? Why is my function not working?

  • $\begingroup$ I am pretty sure that your assumption that the rate of change is linear is what makes this wrong. Charging is a complicated chemical/physical process depending on the forces between atoms and their electrons and between atoms and molecules od different substances whiich makes this question unlinear. $\endgroup$ – Vinyl_cape_jawa Feb 23 at 11:33
  • 3
    $\begingroup$ I'd suggest testing the charging process from 0 to 100%, check the charge every 10 minutes, make a graph afterwards and see how linear it is. Then you can choose your assumptions. $\endgroup$ – Hugh Feb 23 at 11:36
  • $\begingroup$ @Vinyl_coat_jawa I figured. I was hoping I could constrain nonlinear factors and use conventional math to get conventional answers. I thought inputting the actual real-life values would, at least, work even if its rate doesn't work for other values. $\endgroup$ – Lex_i Feb 23 at 11:40
  • $\begingroup$ @Hugh, that's a good idea, lol. I'll try that when I find the time $\endgroup$ – Lex_i Feb 23 at 11:42
  • $\begingroup$ 0.40/216 = 0.1/54 = 0.01/5.4,: 1% per 5 & a half minutes. Whereforth 0.01/9? $\endgroup$ – William Elliot Feb 23 at 12:12

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