Please answer the following questions using exponential and logarithmic models. 4) A wooden artif...

4. Let the initial amount of Carbon be N₀

Amount of carbon after t years = N₀ e^-kt

Given, half life = 5730 years

So, N₀/2 = N₀ e^-k(5730)

=> 1/2 = e^-5730k

=> ln (e^-5730k) = ln (1/2)

=> -5730k = ln(1/2)

=> k = 0.000121

Let the artifact be t years old

0.7N₀ = N₀ e^-0.000121t

=> 0.7 = e^-0.000121t

=> ln (e^-0.000121t) = ln 0.7

=> -0.000121t = ln (0.7)

=> t = 2947.73 years

5. 1-day decay percent = 1.15% = 0.0115

For x percent decrease , decay factor g is given by

g = 1 - 0.0115 = 0.9885

Initial amount = 0.5 grams

Amount of Iodine after t days = Initial Amount * (1-day decay factor)^t

=> A(t) = 0.5(0.9885)^t

Amount of Iodine after 60 days = 0.5(0.9885)⁶⁰ = 0.2498 gm

6. 1-hour decay percent = 20% = 0.2

  For x percent decrease , decay factor g is given by

g = 1 - 0.2 = 0.8

Let the initial amount be N_o

Amount of drug after t hours = N_o (0.8)^t

Let the half life be n hours

So, N_o/2 = N_o (0.8)ⁿ

=> (0.8)ⁿ = 0.5

=> ln (0.8)ⁿ = ln 0.5

=> n ln 0.8 = ln 0.5

=> n = 3 hours

So, the half life of the drug is 3 hours