The problem describes an exponential decay of a substance over time. The table shows measurements of the remaining mass of the substance at different times, but the initial amount is unknown. Since exponential decay follows the model:

$M(t) = M_0 \cdot e^{-kt}$

Where:

• $M(t)$ is the mass at time $t$,
• $M_0$ is the initial mass (what we're solving for),
• $k$ is the decay constant,
• $t$ is time,
• $e$ is the base of the natural logarithm.

Step-by-Step Solution

1. Use the data points from the table to determine the decay constant $k$, then use it to calculate the initial mass $M_0$.

Let's choose two points from the table for simplicity:

• At $t = 4$ hours, $M(4) = 34.5$ g,
• At $t = 5$ hours, $M(5) = 21.7$ g.

We can set up the equation:

$\frac{M(5)}{M(4)} = e^{-k \cdot (5 - 4)}$

$\frac{21.7}{34.5} = e^{-k}$

2. Solve for $k$: $\frac{21.7}{34.5} = 0.629 \quad \Rightarrow \quad e^{-k} = 0.629$

$-k = \ln(0.629) \quad \Rightarrow \quad k = -\ln(0.629)$

$k \approx 0.464$

3. Now, use this $k$ to find $M_0$. Using the first data point ($t = 4$, $M(4) = 34.5$ g):

$M(4) = M_0 \cdot e^{-0.464 \cdot 4}$

$34.5 = M_0 \cdot e^{-1.856}$

$34.5 = M_0 \cdot 0.156$

$M_0 \approx \frac{34.5}{0.156} \approx 221.15 \, \text{g}$

So, the initial mass of the substance is approximately 221 grams.

Would you like to go deeper into any part of this process?

Here are 5 related questions to explore this further:

1. How would the solution change if the decay constant were different?
2. Can you derive the decay constant $k$ using other time points from the table?
3. How would the model change if the decay followed a linear rather than exponential pattern?
4. How long will it take for the substance to reach 1 gram based on this decay rate?
5. What real-world scenarios could this exponential decay model be used for?

Tip: Exponential decay is commonly seen in radioactive decay and population dynamics. It always involves a proportional decrease over time!