Question

a sample contains 800 mg of this isotope. combine the first two equations and laws of exponents to find the equation for the mass of sodium remaining after y years.
s = 800(0.165)^y
s = 800(0.766)^{2.6y}
s = 800(0.5)^{2.6y}
s = 800(0.766)^y

Explanation:

Step1: Recall radioactive - decay formula

The general formula for radioactive decay is $S = S_0\cdot a^{\frac{y}{t_{1/2}}}$, where $S_0$ is the initial amount, $a$ is the decay - factor, $y$ is the time elapsed, and $t_{1/2}$ is the half - life. In the case of exponential decay, if the decay is modeled as $S = S_0\cdot b^y$, we need to find the appropriate decay factor $b$.

Step2: Analyze the problem

We assume a first - order decay process. Without knowing the half - life information explicitly in the problem statement, we can use the fact that for a radioactive substance with initial amount $S_0 = 800$ mg, the general form of the decay equation is $S=S_0\cdot r^y$, where $r$ is the fraction of the substance remaining after one year.
If we assume a non - half - life based approach and just consider the general exponential decay model $S = S_0\cdot r^y$. We need to find the correct value of $r$.
Let's assume that the decay is modeled correctly in one of the given equations. We know that the amount of a radioactive substance $S$ after time $y$ years with initial amount $S_0$ follows an exponential decay model.
If we assume that the decay is such that the amount of sodium remaining $S$ after $y$ years with initial amount $S_0 = 800$ mg is given by $S = S_0\cdot r^y$.
We need to find the correct value of $r$. Usually, in radioactive decay problems, we know that the amount of a substance decreases over time.
If we assume that the decay is modeled as $S = 800\cdot r^y$, we need to find the correct $r$.
For radioactive decay, the amount of a substance $S$ after time $y$ is given by $S = S_0\cdot(0.5)^{\frac{y}{t_{1/2}}}$. But if we assume a non - half - life based exponential decay model $S=S_0\cdot r^y$.
We know that the amount of the substance decreases over time. Among the given options, we assume that the decay factor $r$ is such that the amount of sodium remaining $S$ after $y$ years is $S = 800(0.766)^y$. This is because the general form of exponential decay is $S = S_0\cdot r^y$ where $0\lt r\lt1$.

Answer:

$S = 800(0.766)^y$