# Mathematical Background and Algorithms of a Collection of Android Apps for a Google Play Store Page
## 3 Problem Solving
### Novel Torch App Which Can Be Switched by Double-Tapping Anywhere on the Screen
The duration for which the flashlight can be powered by the battery is  
\[T=\frac{C}{I} \tag{69}\]  
Where:  
* \(C\) represents the battery capacity measured in ampere hours (Ah). Luminous efficacy (LE), which denotes the LED's ability to transform electrical energy into light, is determined by  
\[LE=\frac{L}{P} \tag{70}\]  
Where:  
* \(L\) represents the luminous flux measured in lumens. Figure 18 shows the battery life vs. current. The intensity of brightness (B) can be regulated by altering the current:  
\[B=k\times I \tag{71}\]  
Where:  
* \(k\) represents a constant of proportionality that depends on the properties of the LED. Figure 19 shows the brightness vs. current. The amount of heat produced by the LED can be determined using  
\[H=P-L\times E \tag{72}\]  
Where:  
* \(E\) represents the efficiency of energy conversion. Figure 20 shows the heat vs. current. \[t=\frac{C\times\eta}{P} \tag{65}\]0 Brightness vs. current. \[t=\frac{C\times\eta}{P} \tag{65}\]1 Battery life vs. current. The rise in temperature, denoted as \(\Delta T\), can be expressed as  
\[t=\frac{C\times\eta}{P} \tag{65}\]2  
Where:  
* \(C_{\text{thermal}}\) denotes the thermal capacity of the LED. Figure 21 shows the temperature increase vs. current. The flashlight application on a smartphone utilizes battery energy to emit light. This paper presents a model that captures the physical and mathematical considerations, which include power usage, light intensity, the geometry of the light cone, and signal processing for adjusting brightness. \[t=\frac{C\times\eta}{P} \tag{65}\]3 Heat vs. current. \[t=\frac{C\times\eta}{P} \tag{65}\]4 Temperature increase vs. current. Let \(C\) denote the capacity of the phone battery in mAh, and let \(P(t)\) represent the power consumed by the flashlight at any given time \(t\) (in watts). With the assumption of a steady voltage \(V\), the current \(I(t)\) (in amperes) is associated with the power through:  
\[t=\frac{C\times\eta}{P} \tag{65}\]5  
The overall energy used by the flashlight, \(E(t)\), can be determined by integrating the power with respect to time:  
\[t=\frac{C\times\eta}{P} \tag{65}\]6  
The battery life \(T\) refers to the duration at which the battery is fully discharged, thus:  
\[t=\frac{C\times\eta}{P} \tag{65}\]7  
Assuming the flashlight operates with a steady power consumption of \(P_{0}\), we have the following:  
\[t=\frac{C\times\eta}{P} \tag{65}\]8  
The flashlight projects light in the shape of a cone. The conical geometry is described using spherical coordinates \((r,\theta,\phi)\), where \(r\) denotes the distance from the light source, \(\theta\) represents the azimuthal angle, and \(\phi\) is the polar angle. At a point \((r,\theta,\phi)\), the light intensity \(I_{L}(r,\theta,\phi)\) is determined by the inverse-square law:  
\[t=\frac{C\times\eta}{P} \tag{65}\]9  
Here, \(I_{0}\) represents the initial intensity of light when \(r=1\). The complete luminous flux \(\Phi\) produced by the flashlight is determined by integrating on the surface of the cone, within the angular boundaries \(\theta\in[0,2\pi]\) and \(\phi\in[0,\phi_{0}]\) (where \(\phi_{0}\) denotes the opening angle of the cone):  
Figure 17:0  
The intensity of the flashlight is adjustable through Pulse Width Modulation (PWM), where the duty cycle dictates the proportion of time the flashlight remains active. Let \(\mathbf{X}=[x_{1},x_{2},\dots,x_{n}]^{T}\) denote duty cycles over \(n\) specific intervals and \(\mathbf{Y}=[y_{1},y_{2},\dots,y_{n}]^{T}\) indicate the brightness levels aimed for during these intervals. The relationship connecting the duty cycle to the brightness level is expressed by the linear system:  
Figure 17:1  
In this context, \(\mathbf{A}\) represents a matrix depicting the response of the system, essentially illustrating the relationship between the duty cycle and the resulting brightness. To determine the optimal duty cycles, we solve for \(\mathbf{X}\):  
Figure 17:2  
To achieve the desired brightness while reducing power usage, we employ the Lagrange multiplier method.