PRE_CALC question plzz help

Question

an olympic athlete ingests an illegal substance at 10:00 pm Friday evening. The amount of the substance in the athlete's body after t hours is determined by the function f(t)=100-30ln(1+t). If t=0 represents 10:00 pm Friday, what t value represents 11:00 am saturday morning. if the amount of a n illegal substance in an athlete's body is in excess of 20mL, he will be disqualified. If a drug test is administered at 11:00 am saturday, will the athlete be disqualified and when will he not get disqualified? There will be a similar question like this on a quiz. Please tell me the steps.

Answer 1

f(t) = 100 -30ln(1+t) = amount of substance t hours after 10 PM                                       on Friday

 

On Saturday at 11 AM, t = 13

 

f(13) = 100 - 30ln(14) = 20.828 ml, so he will be disqualified.

 

To avoid disqualification, the athlete needs to delay testing until f(t) ≤ 20.

 

100 - 30ln(1+t) ≤ 20

      - 30ln(1 + t) ≤ -80

            ln(1 + t) ≥ 2.6666666

               1 + t ≥ e^2.6666666

               1 + t ≥ 14.3919

                     t ≥ 13.3919 hours after Friday at 10 PM

                     t ≥ 13hours23.514minutes after Fri at 10 PM

 

To play it safe, delay testing to about 11:24 AM (or later) on Saturday to be sure that the amount of drug remaining is below 20 ml.   For example, at 11:24 AM on Saturday, t = 13.4 hours, so, f(13.4) = 100 - 30ln(14.4) = 19.98 ml, so the athlete would avoid disqualification if testing takes place at that time.