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

Explanation:

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