A researcher models the depth of a sediment layer, D(p)D(p)D(p), in millimetres as a function of the pressure ppp in kilopascals using the modulus function:
D(p)=∣cp−30∣−6,p≥0 D(p) = |cp - 30| - 6, \quad p \ge 0 D(p)=∣cp−30∣−6,p≥0where ccc is a positive constant representing the compression coefficient.
The graph of y=D(p)y = D(p)y=D(p) has a minimum point MMM and intersects the yyy-axis at the point SSS.
(i) Find the yyy-coordinate of SSS. (ii) Find, in terms of ccc, the ppp-coordinate of MMM.
Find, in terms of ccc, the set of values of ppp for which the sediment depth is negative.
A second model suggests the depth follows the linear relationship y=18−4py = 18 - 4py=18−4p. Given that this linear model intersects the modulus model y=D(p)y = D(p)y=D(p) at two distinct points,
find the range of possible values for ccc.
