The problem presented in the image describes a scenario where a radioactive material decays, and you're tasked with determining its half-life. Here's a summary of the information provided:

1. Initial mass of the radioactive material: 2 grams.
2. After 10 days, only 0.1 grams of the material remains.
3. You're asked to determine the half-life of the substance.
4. The hint suggests using the exponential decay model of the form $y = A e^{-kt}$.

Steps to solve the problem:

1. Model setup: The general exponential decay formula is: $y = A e^{-kt}$ where:

   • $y$ is the remaining amount of substance after time $t$,
   • $A$ is the initial amount of the substance,
   • $k$ is the decay constant,
   • $t$ is the time that has passed.

2. Substitute known values:

   • Initially, $A = 2$ grams.
   • After 10 days, $y = 0.1$ grams. $0.1 = 2 e^{-10k}$

3. Solve for $k$: $\frac{0.1}{2} = e^{-10k} \implies 0.05 = e^{-10k}$ Take the natural logarithm on both sides: $\ln(0.05) = -10k$ $k = -\frac{\ln(0.05)}{10}$ Using $\ln(0.05) \approx -2.9957$: $k = \frac{2.9957}{10} = 0.29957$

4. Determine the half-life: The half-life $T_{1/2}$ is the time it takes for half the material to decay. For exponential decay, the half-life is given by: $T_{1/2} = \frac{\ln(2)}{k}$ Using $\ln(2) \approx 0.6931$: $T_{1/2} = \frac{0.6931}{0.29957} \approx 2.314 \, \text{days}$

Answer:

The half-life of the material is approximately 2.31 days (rounded to three significant digits).

Would you like further details or have any other questions?

Here are some related questions to consider:

1. How is the decay constant related to the rate of radioactive decay?
2. What would happen to the decay equation if the initial sample was larger?
3. Can the exponential decay formula be used for other types of decay (like population or charge)?
4. How do you experimentally determine the remaining amount of a radioactive substance?
5. What happens when the time $t$ approaches infinity in the decay model?

Tip:

When solving for a decay constant or half-life, always make sure to check the units of time carefully to ensure consistency throughout the calculation!