Answer:
193 mg
Step-by-step explanation:
Exponential decay formula:
- [tex]A_t = A_0e^r^t[/tex]
- where Aₜ = mass at time t, A₀ = mass at time 0, r = decay constant (rate), t = time
Our known variables are:
- 1998 to the year 2004 is a total of t = 6 years.
- The sample of radioactive isotope has an initial mass of A₀ = 360 mg at time 0 and a mass of Aₜ = 270 mg at time t.
Let's solve for the decay constant of this sample.
- [tex]270=360e^-^r^(^6^)[/tex]
- [tex]270=360e^-^6^r[/tex]
- [tex]\frac{3}{4} =e^-^6^r[/tex]
- [tex]\text{ln} (\frac{3}{4} )= \text{ln}(e^-^6^r)[/tex]
- [tex]\text{ln} (\frac{3}{4} )=-6r[/tex]
- [tex]r=-\frac{\text{ln}\frac{3}{4} }{6}[/tex]
- [tex]r=0.04794701[/tex]
Using our new variables, we can now solve for Aₜ at t = 7 years, since we go from 2004 to 2011.
Our new initial mass is A₀ = 270 mg. We solved for the decay constant, r = 0.04794701.
- [tex]A_t=270e^-^(^0^.^0^4^7^9^4^8^0^1^)^(^7^)[/tex]
- [tex]A_t=270e^-^0^.^3^3^5^6^2^9^0^7[/tex]
- [tex]A_t=193.01982213[/tex]
The expected mass of the sample in the year 2011 would be 193 mg.
