10. [Maximum mark: 15]
The following diagram shows a point series (P, Q) on the Cartesian plane.
900
800
B(15, 700)
700
600
500
C(30, 450)
E(60, 400)
-400
A(0, 400)
300
200
100
D(45, 100)
0 10 15 20 25 30 35 40 45 50 55 60 65 70 *
Peter believes that the points follow a sinusoidal model.
(a) Use the points A(0, 400), B(15, 700), D(45, 100), and E(60, 400) to estimate (i) the equation of the central (or principal) axis. (ii) the amplitude (iii) the period of the model function.
(b) Find a model of the form P = asin(bθ + c) for the model series.
(c) On the same diagram above, sketch the graph of your model.
(d) The exact value of P at Q = 30 is 450. Point C. Write down the estimation of the model for the value of P at θ = 30 and hence find the percentage error.
(e) Peter needs a prediction for the value of P when θ = 90
(i) (ii)
Find an estimation according to your model.
Write down a comment about the validity of your estimation.