This is a problem involving exponential decay. The decay model for a continuous process can be described by the formula:

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

Where:

• $M(t)$ is the mass remaining after time $t$,
• $M_0$ is the initial mass,
• $k$ is the decay constant (relative rate of decay),
• $t$ is the time,
• $e$ is the base of the natural logarithm.

Given:

• $M_0 = 9568$ grams (initial mass),
• $k = 0.15$ (15% per hour decay rate),
• $t = 5$ hours.

We need to find $M(5)$, which is the remaining mass after 5 hours.

Step-by-step Calculation:

1. Apply the values to the exponential decay formula:

$M(5) = 9568 \times e^{-0.15 \times 5}$

2. Simplify the exponent:

$M(5) = 9568 \times e^{-0.75}$

3. Use $e^{-0.75} \approx 0.472$:

$M(5) = 9568 \times 0.472$

4. Multiply:

$M(5) \approx 4516.1 \, \text{grams}$

Thus, the mass of the sample after 5 hours is approximately 4516.1 grams.

Would you like further details or explanations on any step?

Here are 5 related questions for further exploration:

1. How would the remaining mass change if the decay rate was 10% per hour?
2. How can we derive the decay constant if we know the half-life of a substance?
3. How much mass would remain after 10 hours with the same decay rate?
4. What happens to the sample's mass as time approaches infinity?
5. Can you solve a similar problem using a discrete decay model instead of a continuous one?

Tip: Always check whether a decay process is continuous or discrete, as the formulas vary for each model.