# 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...

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~

Thank you!

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### 1 Answer

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)` )

`A'(t)=100(ln1.6)1.6^(-t)(-1)=(-100ln1.6)(1.6)^(-t)~~-47.0004(1.6)^(-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)

`50=100(1.6)^(-t)`

`1.6^(-t)=1/2`

`ln(1.6^(-t))=ln(1/2)`

`t*ln(1.6^(-1))=ln(1/2)`

`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)` :

`A'(1.47)~~-47.0004(1.6)^(-1.47)~~-23.55` units/yr.