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The provided problem involves a chemical reaction with the quantity of reagent, \( Q(t) \), at time \( t \) given by the function:

\[
Q = f(t) = 100e^{t}
\]

The task is to find \( f(4) \), the quantity of the reagent at \( t = 4 \) hours, and \( f'(4) \), the rate of change of the reagent at that same time. Let's break it down:

### Step 1: Calculate \( f(4) \)
The function for the mass of the reagent is \( f(t) = 100e^t \). To find \( f(4) \), we substitute \( t = 4 \) into the equation:

\[
f(4) = 100e^4
\]

### Step 2: Calculate \( f'(t) \)
The derivative of \( f(t) = 100e^t \) is:

\[
f'(t) = 100e^t
\]

Since the exponential function's derivative \( \frac{d}{dt} e^t = e^t \), we have the same expression for the rate of change. To find \( f'(4) \), substitute \( t = 4 \) into the derivative:

\[
f'(4) = 100e^4
\]

I will calculate both values now.The values for the function and its derivative at \( t = 4 \) are:

- \( f(4) = 5459.82 \) mg (rounded to two decimal places)
- \( f'(4) = 5459.82 \) mg/hour

### Interpretation:
- After 4 hours, the quantity of the reagent is approximately 5459.82 mg.
- The quantity of the reagent is increasing at a rate of 5459.82 mg per hour at that time.

Do you have any questions or would you like more details? 

Here are 5 related questions to consider:
1. What happens to the reagent quantity as \( t \to \infty \)?
2. How does the derivative \( f'(t) \) relate to the reaction speed?
3. What would the quantity of reagent be at \( t = 2 \) hours?
4. What is the rate of change of the reagent at \( t = 1 \) hour?
5. How would the results change if the initial mass of the reagent were different?

**Tip:** The exponential function grows very rapidly, especially for larger values of \( t \). Keep this in mind when interpreting the results of the reaction over time.

A chemical processes a reagent whose mass, Q mg, t hours after the reaction takes place is given by Q = f(t) = 100e^t. Find f(4), f'(4) and interpret the result.

Explore how the quantity of a chemical reagent changes over time using exponential growth functions. This advanced problem examines the function Q = f(t) = 100e^t, its derivative, and provides interpretations for specific time values, suitable for college-level students studying calculus.

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