Question

a substance with a half life is decaying exponentially. if there are initially 12 grams of the substance and after 2 hours there are 7 grams, how many grams will remain after 3 hours? round your answer to the nearest hundredth, and do not include units. provide your answer below:

Explanation:

Step1: Write the exponential - decay formula

The general formula for exponential decay is $A = A_0e^{kt}$, where $A_0$ is the initial amount, $A$ is the amount at time $t$, and $k$ is the decay constant. Given $A_0 = 12$, when $t = 2$, $A=7$. Substitute these values into the formula: $7 = 12e^{2k}$.

Step2: Solve for $k$

First, divide both sides of the equation $7 = 12e^{2k}$ by 12: $\frac{7}{12}=e^{2k}$. Then take the natural - logarithm of both sides: $\ln(\frac{7}{12})=\ln(e^{2k})$. Since $\ln(e^{2k}) = 2k$, we have $k=\frac{1}{2}\ln(\frac{7}{12})$. Calculate $\ln(\frac{7}{12})\approx\ln(7)-\ln(12)\approx1.9459 - 2.4849=-0.539$. So $k=\frac{-0.539}{2}=-0.2695$.

Step3: Find the amount at $t = 3$

Now we want to find $A$ when $t = 3$. Use the formula $A = A_0e^{kt}$ with $A_0 = 12$, $k=-0.2695$, and $t = 3$. So $A = 12e^{-0.2695\times3}$. First, calculate $-0.2695\times3=-0.8085$. Then $e^{-0.8085}\approx0.445$. So $A = 12\times0.445 = 5.34$.

Answer:

5.34