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The problem involves modeling the decay of a substance using an exponential decay function. The decay data, which is based on hours and the remaining mass of the substance, is as follows:

| Hours | Mass of substance (g) |
|-------|-----------------------|
| 4     | 34.5                  |
| 5     | 21.7                  |
| 6     | 13.7                  |
| 7     | 8.6                   |
| 8     | 5.4                   |

### Exponential Decay Model
The general form of an exponential decay model is:

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

Where:
- \( M(t) \) is the mass at time \( t \),
- \( M_0 \) is the initial mass (which we are solving for),
- \( k \) is the decay constant (to be determined),
- \( t \) is time in hours.

### Steps to solve:
1. Use the given data points to estimate the decay constant \( k \).
2. Plug one of the data points into the exponential decay equation to estimate \( k \).
3. Use the value of \( k \) to determine \( M_0 \), the initial mass at \( t = 0 \).

I will proceed with calculating these values.The estimated initial mass \( M_0 \) of the substance is approximately **219.72 grams**, and the decay constant \( k \) is about **0.463**.

Would you like further details or clarifications?

Here are five related questions:
1. How does the decay constant \( k \) influence the rate of decay?
2. Can this model be used to predict the time it will take for the substance to fully decay?
3. What is the half-life of the substance, and how can we calculate it using \( k \)?
4. How accurate is this exponential model, and could any other models be more appropriate for certain types of decay?
5. How would an increase in the decay constant affect the shape of the decay curve?

**Tip:** Exponential decay models are commonly used in radioactive decay and chemical reactions, as they describe processes where the rate of change is proportional to the current value.

A certain substance in a laboratory is accidentally exposed to a reagent and starts to decay. 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 estimate the initial mass of a substance undergoing exponential decay. This problem provides a step-by-step approach to using the exponential decay formula, utilizing data points of remaining mass over time, and calculating decay constant. Suitable for students in Grades 10-12.

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