12. f(x) = 10e^{-0.25x} ext{ sin }x, ext{ } x ext{ } extgreater 0 (a) Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation \tan x = 4 (4) (b) Sketch the graph of H against t where H(t) = |10e^{-0.25x} ext{ sin }x| ext{ } | t ext{ } > 0 showing the long-term behaviour of this curve - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 1

To sketch the graph of H(t), we consider the behavior of the function over large values of t. Since the exponential term $e^{-0.25x}$ will lead to H approaching zero as t increases, the long-term behavior is characterized by oscillations of decreasing amplitude.

The graph should have loops that gradually decrease in height, illustrating the damping effect due to the exponential decay.

To find the maximum height, we can evaluate H(t) at the x-coordinates found from the turning points.

Calculating H(4.47):

$H(4.47) = |10e^{-0.25 imes 4.47} ext{ sin }(4.47)|$

Calculating this gives:

$H(4.47) \approx 3.18 ext{ metres}$

Therefore, the maximum height is approximately 3.18 metres.

The model does not take into account various real-life factors that affect a ball's bounce, such as air resistance or energy loss due to impacts.

The heights and timing may vary significantly due to these external factors, making the model unreliable for precise predictions of bounces.