A doctor injected a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there were 4.75 milligrams of dye remaining in the patient system. What is an appropriate model for this situation?
For a decay problem, we will use 𝐴 equals 𝑃 times 𝑒 to the 𝑘𝑡 power. 𝐴 will represent the number of milligrams of dye remaining, 𝑃 will represent how much was injected, 𝑘 will be a constant that we’ll need to find, and 𝑡 will be time in minutes.
So 4.75 milligrams is how much was remaining, 13 milligrams was injected, and 12 minutes went by. So let’s solve for 𝑘. Let’s first divide both sides by 13. Now to get rid of the 𝑒, we need to take the natural log of both sides.
Taking the natural log of both sides cancels out the 𝑒. And now we need to divide both sides by 12. After rounding, 𝑘 is equal to negative 0.0839. So an appropriate model for this situation would be 𝑓 of 𝑡 equals 13 times 𝑒 to the negative 0.839 times 𝑡 power.