The given function is $P(t) = \frac{-3e^{kt}}{k(2 - e^{7k})} + \frac{3}{k}$ with the initial condition $P(0) = 100$.

To solve this problem, we need to find the value of $k$ using the initial condition $P(0) = 100$.

Step-by-step solution:

1. Substitute $t = 0$ into the equation: $P(0) = \frac{-3e^{k \cdot 0}}{k(2 - e^{7k})} + \frac{3}{k}$ Simplifying: $P(0) = \frac{-3 \cdot 1}{k(2 - e^{7k})} + \frac{3}{k}$ So, $P(0) = \frac{-3}{k(2 - e^{7k})} + \frac{3}{k}$

2. Set $P(0) = 100$: $100 = \frac{-3}{k(2 - e^{7k})} + \frac{3}{k}$

3. Solve for $k$: Now we need to solve the equation: $100 = \frac{-3}{k(2 - e^{7k})} + \frac{3}{k}$ Multiply through by $k$ to eliminate the denominators: $100k = \frac{-3}{2 - e^{7k}} + 3$ Rearrange the equation: $100k - 3 = \frac{-3}{2 - e^{7k}}$ $\frac{-3}{100k - 3} = 2 - e^{7k}$ $e^{7k} = 2 + \frac{3}{100k - 3}$ This is the equation that needs to be solved for $k$. At this stage, you would typically use numerical methods to solve this transcendental equation for $k$.

Would you like to go further and compute an approximate value for $k$ using a numerical method?

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Related questions:

1. How can we solve transcendental equations involving exponentials and algebraic terms?
2. What are the numerical methods for solving equations like this one?
3. Can we simplify the equation further before applying a numerical method?
4. What happens if the value of $P(t)$ changes to something else at $t = 0$?
5. How does the behavior of the function $P(t)$ change as $k$ varies?

Tip: When solving equations with exponential functions, numerical solutions are often more practical than algebraic manipulation, especially when the equation cannot be simplified easily.