# Updating half life

You can click the arrows to change the scales of the graph. For example, we fit a linear discrete dynamical system model to the population growth of the bacteria V. The resulting exponential growth equation was $P_T = 0.022 \times 1.032^T$ (equation (6) of the bacteria growth page.) We can plot the V.natriegens along with the model function in a modified version of the above applet. The previous applet shown with data from the population growth of the bacteria V. For the model $P_T = 0.022 \times 1.032^T$ fit to the data, the doubling time is about 22 minutes. Observe that at $T = 26$, $P = 0.05$ and at $T=48$, $P = 0.1$; thus $P$ doubled from 0.05 to 0.1 in the 22 minutes between $T=26$ and $T=48$.We'll show that this $T_$ won't depend on our choice of $t_1$.To determine $t_2$ (and hence $T_$), we must solve the equation \begin x_ = 2 x_ \end which, according to the model of equation \eqref, we can rewrite as \begin x_0 \times b^ = 2 x_0 \times b^.Such exponential growth or decay can be characterized by the time it takes for the population size to double or shrink in half.For exponential growth, we can define a characteristic of a population exhibiting exponential growth is the time required for a population to double.Designed for Microsoft Windows, the game uses a heavily modified version of the Quake engine, called Gold Src.n Half-Life, players assume the role of the protagonist, Dr.  The player takes the perspective of scientist Gordon Freeman.We want to calculate the time $t_2$ at which the population size has double to twice $x_$.If $x_= 2 x_$, then the doubling time is $T_=t_2-t_1$.\end Dividing both sides of the equation by $x_0 \times b^$, we can simplify the condition to \begin \frac = 2, \end which is the same as \begin b^ = 2.\end As we claimed at the beginning, we can find an equation for $T_=t_2-t_1$ that doesn't depend on our choice of $t_1$.