Question

Laplace Transform and Final Value Theorem:

Z(s) = (15s^3 + 2s^2 + 640) / (s(5s^4 + 162s^3 + 1424s^2 + 3744s + 1280))
(a) Find the final value of z(t) using the final value theorem.

Answer 1

The final value of z(t), found using the final value theorem, is 1/2. This is obtained by applying the limit of sZ(s) as s approaches zero after canceling the common factor of s in the numerator and denominator.

Step-by-step explanation:

To find the final value of z(t) using the final value theorem, we apply the theorem which states that if the limits exist, the final value of z(t) as t approaches infinity is given by the limit of s multiplied by Z(s) as s approaches zero:

lim_{t to infty} z(t) = lim_{s to 0} sZ(s)

We have:

Z(s) = {15s^3 + 2s^2 + 640} / {s(5s^4 + 162s^3 + 1424s^2 + 3744s + 1280)}

Substituting 's = 0' into sZ(s) will give us the final value. Since we have a factor of s in the numerator and denominator, they will cancel out. After canceling, we plug in s = 0:

lim_{s to 0} {15s^3 + 2s^2 + 640} / {5s^4 + 162s^3 + 1424s^2 + 3744s + 1280} = lim_{s to 0} 640 / 1280 = 1 / 2

Therefore, the final value of z(t) is 1/2.