AP Calc BC 2023 FRQ #3

3. The depth of seawater at a location can be modeled by the function H that satisfies the differential equation
dH
dt
H
1 t
2
1 cos
2 = ( − ) ( ), where H t( ) is measured in feet and t is measured in hours after noon (t = 0). It is
known that H( )0 = 4.
(a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, y = H t( ),
through the point ( ) 0, 4 .
(b) For 0 is less than t is less than 5, it can be shown that H t( ) is greater than 1. Find the value of t, for 0 is less than t is less than 5, at which H has a critical
point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or
neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your
answer.
(c) Use separation of variables to find y = H(t), the particular solution to the differential equation
dH
dt
H
1 t
2
1 cos
2 = ( − ) ( ) with initial condition H( )0 = 4.