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solve this the same as above but get it correct this time

Let's start by performing the partial fraction decomposition for the given rational function \( F(s) = \frac{7s + 15}{s^3 + 28s^2 + 228s + 576} \). First, we need to factorize the denominator and then proceed with the decomposition. Let me handle this for you.The correct partial fraction decomposition of \( F(s) = \frac{7s + 15}{s^3 + 28s^2 + 228s + 576} \) is:
\[
F(s) = -\frac{97}{100(s + 16)} + \frac{97}{100(s + 6)} - \frac{27}{10(s + 6)^2}
\]

Next, to find the inverse Laplace transform (the time function \( f(t) \)), we will compute the inverse transform of each term separately.It seems I forgot to define the variable \( t \) for the inverse Laplace transformation. Let me fix that and compute the inverse Laplace transform for the time function \( f(t) \).The time function \( f(t) \) that corresponds to the inverse Laplace transform of \( F(s) \) is given by:
\[
f(t) = \frac{(97 - 270t)e^{10t} - 97}{100} e^{-16t}
\]

This expression simplifies the computation and clearly displays how the function behaves over time, factoring in the exponential growth and decay terms.

Would you like more details or have any other questions on this? Here are some follow-up topics you might consider:

1. **Understanding the behavior**: How does \( f(t) \) behave as \( t \) approaches infinity?
2. **Application**: In what contexts might this type of function appear in real-world applications?
3. **Further decomposition**: What is the process of partial fraction decomposition in more detail?
4. **Roots of the denominator**: How would finding the roots of the denominator polynomial affect the complexity of the decomposition?
5. **Laplace transformations**: Can we explore more about the theory behind Laplace transformations?

**Tip**: When dealing with complex rational functions, always simplify your work by factoring the denominator first to make partial fraction decomposition easier.

Discover how to perform partial fraction decomposition on the rational function (7s + 15) / (s^3 + 28s^2 + 228s + 576) and calculate the corresponding inverse Laplace transform. This detailed solution explains each step and simplifies the process, suitable for university-level students in engineering and mathematics.

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