You go to the doctor and he gives you 13 milligrams of radioactive dye. After 12 minutes, 8 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

Question

Physics

Malachi

19 February, 20:13

You go to the doctor and he gives you 13 milligrams of radioactive dye. After 12 minutes, 8 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

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Alicia Oneal · Answer 1

Applying the exponential function of decay

M=Cexp (-kt)

At t = 0 the mass is 13mmg

Therefore

13=Cexp (0)

C=13

M=13exp (-kt)

After 12mins, M=8mmg

8=13exp (-K*12)

8/13=exp (-12k)

0.615=exp (-12k)

Take In of both sides

In (0.615) = -12k

-0.4855=-12k

Then, k=0.0405

Then the equations become

M=13exp (-0.0405t)

We need to find t at M=2mmg

M=13exp (-0.0405t)

2=13exp (-0.0405t)

2/13=exp (-0.0405t)

0.1538=exp (-0.0405t)

Take In of both sides

In (0.1538) = -0.0405t

-1.872=-0.0405t

Then t=-1.82/-0.0405

t=46.22mintes