Darrin drank a latte with 205 milligrams (mg) of caffeine. each hour, the caffeine in darrin’s body diminishes by about
8%.
a. write a formula to model the amount of caffeine remaining in darrin’s system after each hour.
b. write a formula to model the number of hours since darrin drank his latte based on the amount of caffeine in
darrin’s system.
c. use your equation in part (b) to find how long it takes for the caffeine in darrin’s system to drop below
50 mg

a. C(t)=205*(1-0.08)^t

b. t=log_0.92(C(t)/205)=(log_10(C(t)/205))/(log_10(0.92))

c. 16.92 hours

Step-by-step explanation:

Let's say that C(t) is the expression of the amount of caffeine remaining in Darrin's system after t time, hours in this particular case.

a. Then for the first hour the expression would be:

C(t)=205*(1-0.08)

For the second hour:

C(t)=205*(1-0.08)-205*(1-0.08)*(1-0.08)

For the third

C(t)=205*(1-0.08)-205*(1-0.08)*(1-0.08)-205*(1-0.08)*(1-0.08)*(1-0.08)

And so on, for that reason the best way to fit the expression is:

C(t)=205*(1-0.08)^t

2. To find the correct expression for time, we must solve for t the equation recently written above:

Considering that  log_b(a)=c  and  log_b⁡(a)=log_c⁡(a)/log_c⁡(b), then:

t=log_0.92(C(t)/205)

t= (log_10(C(t)/205))/(log_10(0.92))

3. Finally we replace the given value of C(t) into the equation for t:

t= (log_10(50/205))/(log_10(0.92))=16.92

t= 16.92 hours

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