You drink a Starbucks Coffee in SM Seaside with a 120 mg. of caffeine. Each hour, the caffeine in your body system reduces by about 12 percent. How long will you have a 10 mg. of caffeine?

Name: You drink a Starbucks Coffee in SM Seaside with a 120 mg. of caffeine. Each hour, the caffeine in your body system reduces by about 12 percent. How long will you have a 10 mg. of caffeine?
Uploaded: 2022-02-23T17:22:57+00:00
Description: You drink a Starbucks Coffee in SM Seaside with a 120 mg. of caffeine. Each hour, the caffeine in your body system reduces by about 12 percent. How long will you have a 10 mg. of caffeine?

Question

Mary Herrera · Accepted Answer

Given, initially weight of caffeine =120mg.  Each hour, coffeine reduces by 12% i.e.  Each hour remaining caffeine =88%.  0=120mg  1=120×0.88  2=(120×0.88)×0.88  3=((120×0.88)×0.88)×0.88.  t=(((120×0.88)×0.88)×0.88)×…×0.88=120×0.88t  Therefore, we can say that the caffeine debt at time =thrs is:  c(t)=120×(0.88)t  Let, at time t,  c(t)=10mg  ⇒10=120×(0.88)t  ⇒(0.88)t=10120=112  Take log on both sides, we get  log⁡(0.88)t=log⁡(112)  ⇒t×log⁡(0.88)=log⁡(112)  ⇒t=log⁡(112)log⁡(0.88)  =¬{−}1.07918¬{−}0.05552  =19.437hr.  ≈19.5hrs  or 19 hours 30 minutes  Hence, at time t=19 hours 30 minutes remaining caffeine is 10mg.

Melinda McCombs · Accepted Answer

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.  TimeCaffeine Still Present0120mg.1120−120(0.12)2(120−120(0.12))−...  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.  TimeCaffeine Still Present01201120⋅0.882120⋅0.88⋅0.883120⋅0.88⋅0.88⋅0.884120⋅0.88⋅0.88⋅0.88⋅0.885120⋅(0.88)5  C(t)=120⋅(0.88)t, function notation chosen. Amount of caffeine remaining after t number of hours, is C(t). 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(t)=10. You start with this: 10=120⋅(0.88)t and you want to solve for t.  10120=0.88t  112=0.88t  log10112=log100.88t  log10112=t⋅log100.88  t=log⁡112log⁡0.88  t=−1.07918−0.05552  t=19.4 hours=19 hours 30 minutes.

karton · Accepted Answer

Since the amount if Caffeine decreases by 12% an hour, that is the same as a multiplier of 0.88 per hour. r=ln⁡0.88=−0.11783337110=120e−0.117833371t10/120=e−0.117833371t0.8333=10/120ln⁡0.8333=−2.48490665 soe−2.48490665=e−0.117833371t and so−2.48490665=−0.117833371t−2.48490665/−0.117833371=t21.08830995 hrs=t

You drink a Starbucks Coffee in SM Seaside with a

Answered question

Answer & Explanation

New Questions in Differential Equations