12. (a) Show that the $x$ coordinates of the turning points of the curve with equation $y = f(x)$ satisfy the equation \( \tan x = 4 \) (b) Sketch the graph of $H(t)$ where \[ H(t) = |10 e^{-0.25t} \sin t|, \quad t > 0 \] showing the long-term behaviour of this curve - Edexcel - A-Level Maths Pure - Question 13 - 2019 - Paper 1

To sketch the graph of $H(t)$, we note that the function oscillates due to the sine component, while the $e^{-0.25t}$ factor causes the amplitude to decrease over time.

1. The function oscillates between $0$ and $10$ as the maximum value occurs when $ext{sin}(t) = 1$.
2. As $t$ increases, the exponential decay ensures that the height diminishes, leading to a graph that approaches $0$ but never touches it, showing long-term behavior with decreasing heights.

To find the maximum height between the first and second bounce, we first need to solve for $t$ when the height $H(t)$ is maximized. This occurs when: $\frac{dH}{dt} = 0$ which implies finding:

1. Set $H(t) = 10 e^{-0.25t} \sin t$.
2. Solving $\tan t = 4$ gives relevant $t$ values.

Substituting $t = 4.47$ into $H(t)$ gives: $H(4.47) = 10 e^{-0.25 \cdot 4.47} \sin(4.47).$ Calculating gives approximately $3.18$ meters.

The model $H(t) = |10 e^{-0.25t} \sin t|$ assumes ideal conditions where only gravitational forces act on the ball, neglecting any other factors such as air resistance or energy loss during bounces. Additionally, each bounce is not guaranteed to have the same height and thus varying time intervals may occur, making it unreliable for predicting precise timing for bounces.