12. f(x) = 10e^{-0.25x} \, ext{sin} \, x, \, x \, > \, 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^{-0.25t} \, ext{sin} \, t| \, |t| > 0 showing the long-term behaviour of this curve - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 1

To sketch the graph of H against t, we first note that H(t) oscillates due to the sine function while being damped by the exponential function.

1. As t increases, the term $e^{-0.25t}$ decreases, which means the amplitude of the oscillation reduces over time.
2. The behavior of H(t) will exhibit peaks that gradually decrease in height, effectively showing a decaying sinusoidal wave.
3. For the graph, ensure there is a smooth decay in the wave's height, illustrating the long-term behavior accurately.

To find the maximum height of the ball above the ground, we need to maximize H(t). This occurs when $\sin t = 1$, since H(t) is proportional to $\sin t$:

Using:

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

Maximizing gives:

$H(t) = 10e^{-0.25t}$

We can find this maximum conditionally for the first bounce, considering the first maximum occurs approximately at t = 1.33 (from the turning points, solving \tan x = 4 yields a related t).

Thus, substituting this value:

$H(1.33) = 10e^{-0.25 \cdot 1.33} \approx 10e^{-0.3325} \approx 6.96 ext{m}$

Therefore, the maximum height is approximately 6.96 meters.

The model only accounts for the oscillation of height based on sine functions and exponential decay but does not incorporate real-world factors such as energy loss due to air resistance, impact duration, and ground friction. These can cause significant deviations from the predicted model, which assumes ideal bouncing without such losses. Additionally, each bounce duration may vary if the height significantly decreases over time, leading to unpredictability in bounce timings.