12. (a) Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation $\tan x = 4$ - Edexcel - A-Level Maths Pure - Question 14 - 2019 - Paper 1

To sketch the graph of H against t:

• Begin by noting that $H(t) = |10e^{-0.25t} \sin x|$ will fluctuate between 0 and 10 as $\sin x$ oscillates between -1 and 1.
• The factor $e^{-0.25t}$ indicates that as t increases, the height will decay exponentially.
• Therefore, we expect to see two peaks for each period of the sine wave, but the amplitude of the peaks will decrease over time.
• The graph should show the height tapering off towards zero as t goes on.

We can find the maximum height by substituting the value of $x$ that satisfies $an x = 4$. Using a calculator, we find:

$x_1 = \arctan(4) \approx 1.3258 \quad (first\ turning\ point)$

Substituting this into H, we get:

$H(1.3258) = |10e^{-0.25 \cdot 1.3258} \sin(1.3258)|$

Calculating gives:

$\approx 10e^{-0.33145} \sin(1.3258) \approx 3.18 \text{ metres}$

Thus, the maximum height is approximately 3.18 metres.

This model assumes that the conditions remain constant and does not take into account external factors such as air resistance or changes in the initial conditions of the ball's kick. Additionally, as the height decreases over time, the model fails to accurately predict the timings of subsequent bounces as they will become irregular and influenced by unforeseen variables.