Radioactive substance decays so that after t years, the amount remaining, expressed as a percent of the original amount, is A(t)=100(1.6)^(-t).
Determine the function A’, which represent the rate of decay of the substance.
b) what is the half-life for this substance?
c) what is the rate of decay when half the substance has decayed?
Help/explanation appreciated. I have to know this question similiar on my test tomorrow and I have no idea what to do~
Given the equation for the amount of a substance remaining after `t` years as `A(t)=100(1.6)^(-t)` :
(1) Determine the rate of decay of the substance.
We find `A'(t)` : (Use `d/(dt)a^u=ln(a)a^u(du)/(dt)` and `d/(dt)ku(t)=kd/(dt)u(t)` )
(2) Find the half-life of the substance: We need to find `t` such that `A(t)=50` (Note that `A(0)=100` is the initial amount)
`t=(ln(1/2))/(ln1.6^(-1))~~1.47` So the half-life is approximately 1.47 years
(3) We want the rate of decay when half of the substance has decayed or `A'(1.47)` :