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,

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) (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.

-
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_____________
2_____________
3_____________
4_____________
5_____________, you should see the pattern already, and using exponents make the notation more compact.

, function notation chosen. Amount of caffeine remaining after number of hours, is . 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 when .
You start with this: and you want to solve for .

----------------SOLVING FOR t------------

hours = 19 hours 30 minutes
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