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You go to the doctor and he gives you 17 milligrams of radioactive dye. After 16 minutes, 7.75 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.

Answer :

debrubs97

Answer:

44 min

Step-by-step explanation:

We can assume that the radioactive dye will decay exponentially.  

the equation will be [tex]A(t)=A0ex^{-kt}[/tex]  

where k is a positive constant and t the minutes that takes to go to A from A0. In milligrams:

 

[tex]7.75=17ex^{-k16}[/tex]  

 

k =[tex]k= [-ln(7.75/17) / 16] min-1[/tex] ≅ 0.049095 min-1

 

we find the time we need to spent to have 2miligrams of dye remain:

2 = 17e-0.049095t

 and solve for t.

 t = -ln(2/17) / (-0.049095) min ≅ 43.59 min

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