A substance increases by 25% every 5 minutes. If the current population of substance is 400, how long will it take the population to reach 1 million?
Help please!!
The basic equation that you need to work is this:
$\displaystyle 400 \times 1.25^N = 1,000,000 $
where N = the number of 5-minute periods. You can manipulate it as follows:
$\displaystyle 1.25^N = \frac {1,000,000}{400} = 2500$
$\displaystyle \log (1.25^N) = N \log(1.25) = \log(2500)$
$\displaystyle N = \frac {\log(2500)}{\log 1.25}$
Now you have N, and the time required is 5 minutes times that.
"Increases by 25%" means it goes from A to A+ .25A= (1.25)A every 5 minutes. That is, every 5 minutes you multiply by 1.25:
If it starts at 400, in 5 minutes it will be 400(1.25). After 10= 2(5) minutes it will be multiplied by 1.25 again- it will now be $\displaystyle 400(1.25)(1.25)= 400(1.25)^2$. After 15= 3(5) minutes it will be multiplied by 1.25 another time- it will now be $\displaystyle 400(1.25)(1.25)(1.25)= 400(1.25)^3$. In t minutes there will be N= t/5 periods of 5 minutes and so the 400 will have been multiplied by 1.25 t/5 times giving ebaines' $\displaystyle 400(1.25)^N$