A model for the height, H$($t$),$ in cm of a reclining patient$$s chest as they breathe

has the form

H$($t$)=$ A cos$($Bt$)+$ C

where t is measured in seconds. If the maximum height of the patient$$s chest is

$20$ cm$,$ the minimum height is $15$ cm and they complete one full inhale$-$exhale cycle

in $3$ seconds, determine the values of A$,$ B$,$ and C$.$