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 2

To sketch the graph of H(t) = |10e^{-0.25t} sin t| for t > 0:

1. The function consists of an exponentially decaying envelope, modulated by the sine function.
2. Draw the sine wave which oscillates between -1 and 1, and scale it by 10, leading to oscillation between -10 and 10.
3. The absolute value means the oscillation will remain above the x-axis, creating a wave-like pattern that decreases in amplitude over time.
4. Ensure that the graph reflects that the wave's height diminishes due to the exponential decay factor.

To find the maximum height between the first and second bounce, calculate:

1. Solve for t when H(t) is maximized during the time of the first bounce, which occurs at t = 1.33.
2. Substitute t = 1.33 into H:

$H(1.33) = 10e^{-0.25(1.33)} sin(1.33)$

3. Calculate the value:

$H(1.33) ext{ (using calculator)} = ext{approximately } 6.69 ext{ m.}$

Thus, the maximum height is about 6.69 m.

This model primarily takes into account the initial conditions and simplifies the mechanics of the bounces. However, real-world factors such as energy loss during impacts, air resistance, and changes in ball elasticity can influence actual bounce times. The model does not account for variations that may affect the height and timing of each subsequent bounce, hence it cannot reliably predict bounce intervals.