Question

Express the function below using window and step functions and compute its Laplace transform. g(t) 10- 0 0 2 4 6 8 10 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. Express g(t) using window and step functions. Choose the correct answer below. A. g(t)= u(t - 4) + (8t – 32)I14,5(t) + (- 8t+48)/15,6(t) + u(t-6) O B. g(t) = (8t - 4)u(t - 4)+(-8t+6)u(t-6) C. g(t) = (8t – 32)114,6 (t) +(-8t+48)u(t-5) D. g(t) = (8t – 32)114,5(t) + (- 8t+48)I15,6(t) + Compute the Laplace transform of g(t). L{g} = (Type an expression using s as the variable.)

Answer 1

Given (58) first we will find line careations (st-322 -3x+48 lire caration for (4,0) to (5,8) is X2 42 5 6 y_y xe-x, x-xo y = 8x-32 1 X-4 for (5,8) to (6,0) is y-8-) -s(8-5) = 4-8 -8x+ho-y-8 6-5 2-5 y=-8x+48 : g(t) = ostch 8t-32 4<t<5 -8t+48 5 st<6 O t26. tca 1 we know uct-a)= 60 { Tab(t) = (1 astab { 1 tza otherwise act-a) - uCt-6) our glt) can be wotten as Tho(t) i. e (84-32); T15,6 (+) i. e (8748) = g(t) = Tys (t) (81-32) + (-8t+48) T15, 6 Ct). (or) 8(t)- (8t-32) TTh5CH) + 8t+48) 56(t) ges[et_32) [LCE-6) - u ct-s»]] +Es++) (uct-s)- uct--3) OPTION-D

Laplace transform of g (t). w.kot hluct-a)) = -as t( f (t-a) uct-a)= é as 2 (F(t) S L(tn) = n! Sa+! So g (t)=(8t-32) UCt-4)-(&t-32) uct-s) + (-8t+48) ult-s) - (-8t+48) ult-62 Let's take individually L ((st-32) Cuct-4) = 8L C(t-4) u Ct-4) SL C(t-1) uCt-4) = Seks 2 (t) = sets S2 L ((84-32) uct-5) = 8L ((t-4) u (t-s)) = 8h (Ct-4-1+) uct-5) => 82((+-5) u (t-s) + uct-s)) + [:55.400) é 55 s -55 -55 e BE s2 S <((-8t+48) 4Ct-s)) = -SL ((t-6) uct-5) = -81 ((t-suct-5) - UCE-5). ---[265266) - .55 = -8 [e-5s S2 L ((-8t+48) u Ct-6)). -81 ((t-6) uct-6)) - 8 e GS LCt) -se-6s sa

So se 6-65 80-55 + L(8 (t) = 8ets 8e5s - Set 86-55 52 + 8 e5s s s2 S2 se LCg (t)= sets S2 16e-5s + 86-65 s? sa 25 8 é-65 L(g(t) = [es zest 52 short cut octch so as g (t) = 8t-32 hct<5 -8t+48 sctc6 to 6 (68) 45 g(t) = (IT (+) (88-32) + To 5,0CP) 8t+48)] g(t)- [&t=se) [uct-6) - uct-5»]+E-8t+48) Cuct-s) -uct-6)] can be added g (t) = (8t-32) UCE-4) - (8t-32) uct-5) + (-8t+48) ult-5) - (2 8t +48) ult-6) g(t) = (84-32) ult-4) + (-16€ +80) UCt-s)-((-8++48) uct-6)) 8[(t-)uct--) - 2(t-5) uct-5) +-6)uct-6)} -65 8 [ ets + 52 2655 s2 52 g.ct): SE [229_22+] - sees (e-1))