SOLUTION: Please show step by step. Thanks An unknown radioactive element decays into non-radioactive substances. In 980 days the radioactivity of a sample decreases by 52 percent.

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please show step by step. Thanks An unknown radioactive element decays into non-radioactive substances. In 980 days the radioactivity of a sample decreases by 52 percent.       Log On


   

Question 998642: Please show step by step. Thanks

An unknown radioactive element decays into non-radioactive substances. In 980 days the radioactivity of a sample decreases by 52 percent.
(a) What is the half-life of the element?
half-life: (days)
(b) How long will it take for a sample of 100 mg to decay to 77 mg?
time needed
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
General model y=pe%5E%28-kt%29 and you have starter information so that you can find the value for k. Also half-life.

The 52% decrease after 980 days,
y%2Fp=e%5E%28-kt%29
ln%28y%2Fp%29=-kt
ln%28p%2Fy%29=kt-----------------------(and this step you will use again).
highlight%28k=%281%2Ft%29ln%28p%2Fy%29%29
-
Information given in description is system%28t=980%2Adays%2Cp=100%2Cy=48%29, and this "48" is because that is what remains as percent after 52% decayed.

You should be able to do the rest of the questions and steps.

---
Still wanted more steps to the solutions---

(a) Half-Life
Return to this part: ln%28p%2Fy%29=kt
What was k? Substitute the given values in the main part of the description!
k=ln%28100%2F48%29%2F980
highlight%28k=7.489%2A10%5E%28-4%29%29
The model for decay is now highlight_green%28y=pe%5E%28-0.0007489%2At%29%29.
-
The step to find any time t,
t=ln%28p%2Fy%29%2Fk
You know the needed values for finding half-life with this formula, so substitute THEM.
t=ln%281%2F%281%2F2%29%29%2F0.0007489
highlight%28t=925%2Adays%29----the half-life.

(b) How long to decay from 100 mg to 77 mg ?
You have the formula and model to use.