Question

Need help with these problems.

1. A doctor prescribes 150 milligrams of a therapeutic drug that decays by about 10% each hour. Write an exponential model representing the amount A in milligrams of the drug remaining in the patient's system after t hours.

2. Suppose a sample of a radioactive substance weighs 38 mg. One year later, the sample weighs 23.5mg. What is the half-life of this substance?

3. Carbon-14 has a half-life of 5,730 years. A fossil is found that has 12% of the carbon-14 found in a living sample. How old is the fossil?

Answer 1

1) Initial drug amount, Ao = 150 mg

Rate of decay, r = 10/100 each hour = 0.1 mg/hr

Exponential decay model : A = Ao (1-r)^t

   A = 150 mg (1-0.1)^t = 150 x 0.9^t

   for example If t = 3 hrs, then A = 150 x 0.9³ = 150 x 0.729 = 109.35 g

2) Initial radioactive substance, A0 = 38 mg

t = 1 year

   At = 23.5 mg

At = A0 (0.5)^t/h

   h = half life and t = time

   In (A0/At) = kt

In (38/23.5) = k x 1 year

   In 1.617 = k x 1 year

   0.481/1 = k

decay constant, k = 0.481 /year

t_0.5 = 0.693/k

t_0.5 = 0.693/0.481 /year = 1.44 year

3) t_0.5 = 5730 year

   In (A0/At) = kt

k = 0.693/t0.5

In (A0/At) = 0.693/t0.5) t

   In (100/12) = (0.693/5730 years) x t

   In 8.33 = 0.000121 x t

   2.1198/0.000121 = t

t = 17,519 years old