Final Answer:
The inverse Laplace transform of the given function (9/(s(s² + 16))) using the convolution theorem is sin(4t).
Step-by-step explanation:
In the convolution theorem, the Laplace transform of a product of two functions in the time domain is equal to the convolution of their individual Laplace transforms. In this case, the given function can be expressed as the product of two functions: 9/s and 1/(s² + 16).
The Laplace transform of 9/s is 9, and the Laplace transform of 1/(s² + 16) is (1/4) * sin(4t). Therefore, according to the convolution theorem, the inverse Laplace transform of the given function is the convolution of these individual inverse Laplace transforms, which simplifies to sin(4t).
This result can be verified by taking the Laplace transform of sin(4t), which indeed matches the original expression, confirming the correctness of the inverse Laplace transform using the convolution theorem. The convolution theorem provides an efficient method for finding the inverse Laplace transform of product functions, simplifying complex calculations into more manageable steps.