Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable model for the data. Use Jane’s model to

(i)     write down the height of the cliff;

(ii)    find the height of the rock after 4.5 seconds;

(iii)   find after how many seconds the height of the rock is \(30{\text{ m}}\).

Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for the data. Use Kevin’s model to find when the rock hits the ground.

(i)     On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, plot the data given in the table.

(ii)    By comparing the graphs of f and g with the plotted data, explain which function is a better model for the height of the falling rock.

(i) \(106{\text{ m}}\)     A1     N1

(ii) substitute \(t = 4.5\)     M1

\(h = 44.9{\text{ m}}\)     A1     N2

(iii) set up suitable equation     M1

e.g. \(f(t) = 30\)

\(t = 4.91\)     A1     N1

[5 marks]

recognizing that height is 0     A1

set up suitable equation     M1

e.g. \(g(t) = 0\)

\(t = 5.39{\text{ secs}}\)     A1     N2

[3 marks]

     A1A2     N3

Note: Award A1 for correct scales on axes, A2 for 5 correct points, A1 for 3 or 4 correct points.

(ii) Jane’s function, with 2 valid reasons     A1R1R1     N3

e.g. Jane’s passes very close to all the points, Kevin’s has the rock clearly going up initially – not possible if rock falls

Note: Although Jane’s also goes up initially, it only goes up very slightly, and so is the better model.

[6 marks]