Final answer:
The Laplace transform of the given function can be found by splitting it into two terms and applying the formula for the Laplace transform of each term separately. Then, we combine the two Laplace transforms to find the final result.
Step-by-step explanation:
The Laplace transform of the function f(t) = (5 - t)(h(t - 5) - h(t - 9)) can be found using the formula for the Laplace transform of a piecewise function. Since h(t) is the Heaviside step function, we can split the function f(t) into two separate terms:
f(t) = (5 - t)h(t - 5) - (5 - t)h(t - 9)
Now, we can apply the formula for the Laplace transform of each term separately:
L{(5 - t)h(t - 5)} = 1/s^2 - 5/s^3
L{(5 - t)h(t - 9)} = 1/s^2 - 9e^(-9s)/s^3
Finally, we can combine these two Laplace transforms to find F(s):
F(s) = L{f(t)} = 1/s^2 - 5/s^3 - 1/s^2 + 9e^(-9s)/s^3