Question

A sample of a radioactive isotope had an initial mass of 110 mg in the year 1990 and decays exponentially over time. A measurement in the year 1993 found that the sample's mass had decayed to 90mg. What would be the expected mass of the sample in the year 2002, to the nearest whole number?

Answer 1

Answer:

  49 mg

Step-by-step explanation:

You want the mass of an isotope in 2002 if it was 110 mg in 1990 and decayed to 90 mg in 1993.

Exponential function

The exponential function modeling the remaining amount (y) after t years can be written as ...

  y = (initial amount) × (decay factor)^(t/(decay period))

where the decay factor is the multiplier of the initial amount after a time equal to the decay period.

Application

Here, we have ...

  initial amount = 110 mg
  decay factor = 90/1110
  decay period = 3 years . . . . from 1990 to 1993

This means we can write the function describing the remaining amount as ...

  $y=110\left(\dfrac{90}{110}\right)^{\dfrac{t}{3}}$

After the 12 years from 1990 to 2002, the amount remaining is ...

  $y=(110\text{ mg})\left(\dfrac{90}{110}\right)^{\dfrac{12}{3}}\approx49\text{ mg}$

The mass of the remaining isotope in the year 2002 is about 49 mg.