A certain substance in a laboratory is accidentally exposed to a reagent and starts to decay. However, this goes on for some time before it is noticed. Once the lab scientists notice what is happening, they decide to measure the remainder of this phenomenon until the substance has completely decayed. 
Hours	Mass of
substance (g)
4	34.5
5	21.7
6	13.7
7	8.6
8	5.4

This type of decay is known to follow an exponential curve. About how many grams of the substance were initially exposed to the reagent?

To determine the initial amount of the substance exposed to the reagent, we need to model the decay using an exponential function.

The general form of an exponential decay function is:

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

Where:
- \( M(t) \) is the mass of the substance at time \( t \),
- \( M_0 \) is the initial mass of the substance,
- \( k \) is the decay constant,
- \( t \) is the time in hours.

Given the data, we can use two points to find \( M_0 \) and \( k \).

### Step 1: Set Up the Equations

Let's use the data points at \( t = 4 \) hours and \( t = 5 \) hours:

\[
M(4) = M_0 \cdot e^{-4k} = 34.5
\]
\[
M(5) = M_0 \cdot e^{-5k} = 21.7
\]

### Step 2: Divide the Equations to Eliminate \( M_0 \)

\[
\frac{M(5)}{M(4)} = \frac{M_0 \cdot e^{-5k}}{M_0 \cdot e^{-4k}} = e^{-k} = \frac{21.7}{34.5}
\]

\[
e^{-k} = \frac{21.7}{34.5} \approx 0.629
\]

### Step 3: Solve for \( k \)

Take the natural logarithm of both sides to solve for \( k \):

\[
-k = \ln(0.629)
\]

\[
k \approx -\ln(0.629) \approx 0.463
\]

### Step 4: Substitute \( k \) Back to Find \( M_0 \)

Using the value of \( k \), substitute it back into the equation for \( t = 4 \):

\[
34.5 = M_0 \cdot e^{-4(0.463)}
\]

\[
34.5 = M_0 \cdot e^{-1.852}
\]

\[
M_0 = \frac{34.5}{e^{-1.852}} = 34.5 \times e^{1.852} \approx 34.5 \times 6.37 \approx 219.6 \, \text{g}
\]

### Conclusion

The initial mass of the substance that was exposed to the reagent is approximately **219.6 grams**.

Would you like any further details or have any questions?

---

Here are 5 related questions for further exploration:

1. How does changing the decay constant \( k \) affect the decay rate of the substance?
2. Can the decay process be reversed or slowed down in any way?
3. What is the half-life of this substance based on the decay constant \( k \)?
4. How would the model change if the substance exhibited linear decay instead of exponential decay?
5. What are some real-world examples of substances that follow an exponential decay model?

**Tip:** Exponential decay is common in radioactive decay, cooling processes, and certain types of chemical reactions. Understanding the decay constant can give insights into how quickly these processes occur.

A certain substance in a laboratory is accidentally exposed to a reagent and starts to decay. However, this goes on for some time before it is noticed. Once the lab scientists notice what is happening, they decide to measure the remainder of this phenomenon until the substance has completely decayed. Hours | Mass of substance (g): 4 | 34.5, 5 | 21.7, 6 | 13.7, 7 | 8.6, 8 | 5.4. This type of decay is known to follow an exponential curve. About how many grams of the substance were initially exposed to the reagent?

Learn how to determine the initial mass of a substance in a laboratory setting that follows exponential decay. This problem uses data points measured over several hours to apply the exponential decay formula and natural logarithms, ideal for students in Grades 10-12.

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