SOLUTION: "You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by 12%. How long until you have 10 mg of caffeine?" Hello thank you for helping me,

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: "You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by 12%. How long until you have 10 mg of caffeine?" Hello thank you for helping me,       Log On


   

Question 839175: "You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by 12%. How long until you have 10 mg of caffeine?" Hello thank you for helping me, I do not know where to start with this question. I know that I can use a chart but my teacher said to solve this with an exponent instead of taking 12% repeatedly. Something like 12^x and multiplying.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Look at this problem, how you say, taking 12% repeatedly; but LOOK FOR THE PATTERN. You can symbolize that pattern and put it into an equation.

Time________________Caffeine Still Present
0___________________120 mg.
1___________________120-120(0.12)
2___________________(120-120(0.12))-......
I see what you mean.


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Try from another point of view. The amount REMAINING after each hour is 100-12 percent; and this percentage is applied each hour.
100-12=88.
Instead of examining how 12% of the previous caffeine is eliminated each hour, look at the 88% of caffeine which still remains after every hour.

Time__________Ceffeine Still Present
0_____________120
1_____________120%2A0.88
2_____________120%2A0.88%2A0.88
3_____________120%2A0.88%2A0.88%2A0.88
4_____________120%2A0.88%2A0.88%2A0.88%2A0.88
5_____________120%280.88%29%5E5, you should see the pattern already, and using exponents make the notation more compact.

highlight%28C%28t%29=120%280.88%29%5Et%29, function notation chosen. Amount of caffeine remaining after t number of hours, is C%28t%29. This is exponential decay, the rate being loss of 12% each hour. Initial amount at time zero is 120 milligrams.

Your question then becomes, find t when C%28t%29=10.
You start with this: highlight%2810=120%280.88%29%5Et%29 and you want to solve for t.


----------------SOLVING FOR t------------
10%2F120=0.88%5Et
1%2F12=0.88t
log%2810%2C1%2F12%29=log%2810%2C0.88%5Et%29
log%2810%2C1%2F12%29=t%2Alog%2810%2C0.88%29
t=log%281%2F12%29%2Flog%280.88%29
t=%28-1.07918%29%2F%28-0.05552%29
highlight%28t=19.4%29 hours = 19 hours 30 minutes