Question

A sample of a radioactive isotope had an initial mass of 360 mg in the year 1998 and

Answer 1

Answer:

193 mg

Step-by-step explanation:

Exponential decay formula:

• $A_t = A_0e^r^t$
• 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.

• $270=360e^-^r^(^6^)$
• $270=360e^-^6^r$
• $\frac{3}{4} =e^-^6^r$
• $\text{ln} (\frac{3}{4} )= \text{ln}(e^-^6^r)$
• $\text{ln} (\frac{3}{4} )=-6r$
• $r=-\frac{\text{ln}\frac{3}{4} }{6}$
• $r=0.04794701$

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.

• $A_t=270e^-^(^0^.^0^4^7^9^4^8^0^1^)^(^7^)$
• $A_t=270e^-^0^.^3^3^5^6^2^9^0^7$
• $A_t=193.01982213$

The expected mass of the sample in the year 2011 would be 193 mg.