The vertical displacement of an SUV's body modelled by h(t) = e^(-0.5t) sint where h is the vertical displacement (m) at time (s). Determine when & what the max displacement of the vehicle's body is

Question

when displacement occurs and what it is?

Answer 1

Answer 2

h=e^(-t/2)sin(t)

ln(h)=(-t/2)+ln(sin(t))

h'/h=-(1/2)+cot(t)

h'=(e^(-t/2)sin(t))*(cot(t)-1/2)

h'=e^(-t/2)(cos(t)-(1/2)sin(t))

h'=0 when t=tan^-1(2)+nπ,,,,,,t>0 ,,,, h(t) term of e^(-t/2) implies t=tan^-1(2) is time of maximum displacement

h(tan^-1(2))=e^{((-arctan(2))/2)}*(2/√5)