Answer
(a) $tH(t) = 0~~~~~$ if $t \lt 0$
$tH(t) = t~~~~~$ if $t \geq 0$
We can see a sketch of the graph $y = tH(t)$ below.
(b) $V(t) = 2tH(t)$
We can see a sketch of the graph below.
(c) $V(t) = 4(t-7)H(t-7)$
We can see a sketch of the graph below.
Work Step by Step
(a) $tH(t) = 0~~~~~$ if $t \lt 0$
$tH(t) = t~~~~~$ if $t \geq 0$
We can see a sketch of the graph $y = tH(t)$ below.
(b) If the voltage is gradually turned on and reaches a level of 120 V in a time of 60 seconds, then $c=2$ in the ramp function.
$V(t) = 2tH(t)$
We can see a sketch of the graph below.
(c) If the voltage is gradually turned on and reaches a level of 100 V in a time of 25 seconds, then $c=4$ in the ramp function.
We also need to include $(t-7)$ to account for the 7 second delay. Thus the graph is shifted horizontally to the right by 7 units.
$V(t) = 4(t-7)H(t-7)$
We can see a sketch of the graph below.
