Pre-Calc Question on Exponential Decay

Question

The most common use of radioactive isotopes in medicine is to diagnose disease. Typically, radioactive materials are injected into the body and then radiation-detecting cameras are used to build images of affected body parts or particular types of cells.The equation shown models the amount of a radioactive isotope y(t) in the bloodstream t minutes after it is injected. y=y0e^-.005t

b. What is the rate of growth or decay?

c. What do you think the constant y0 represent in this model? d. The isotope is no longer useful in diagnostic tests when 95% of the sample has decayed. Determine how long a sample will be useful in a diagnostic lab.

Answer 1

The rate of decay is .005

yO represents the initial amount of radioactive material

If 95% of the sample has decayed then only 5% remains. If we started with 100 grams then we are looking for when only 5 grams are left so

5= 100e^(-005t)

.05 = e^(-.005t) Now take the natural log of both sides which gives us

ln.05 = -.005t

t= ln.05/-(.005)

t=599.146 years

Answer 2