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.
Practise AQA A Level Maths 1.5 B: Algebra and functions with exam-style questions for A Level Maths. 358 questions covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
