Question

You go to the doctor and he gives you 10 milligrams of radioactive dye. After 16 minutes, 6.25 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.

You will spend minutes at the doctor's office.

Answer 1

The time it will take you to visit the doctor is: 45 minutes

How to solve exponential word problems?

apply the exponential function:

A = P
`e^(rt)`

You go to the doctor and he gives you 10 milligrams of radioactive dye. After 12 minutes, 6.5 milligrams of dye remain in your system.

we can find r:

6.5 = 10
`e^(r*12)`

6.5/10 =
`e^(12r)`

0.65 =
`e^(12r)`

ln(0.65) = 12r

ln(0.65)/12 = r

Our equation is now:

A = 10
`e^(t*ln(0.65)/12)`

2 = 10
`e^(t*ln(0.65)/12)`

0.2 = e^{t*ln(0.65)/12}

ln(0.2) = t*ln(0.65)/12

12ln(0.2) = t*ln(0.65)

12ln(0.2)/ln(0.65) = t

44.83 = t

rounding:

45 minutes = t