Question

The value of a dollar decreases by 23% every year. Find the half-life in this situation. The half-life is years. (Do not round until the final answer. Then round to the nearest tenth as needed.)

Answer 1

Let y(t) be the value of dollar at time t. We are given that the value of dollar decreases by 23% every year Hence, the rate is r = 0.23 Hence, we have dy -0.23y dt dy -0.23 dt y SE- 0.2 0.23 dt = In ly = -0.23.t +C where C is integration constant Since y > 0 + In ly = ln y → Iny = -0.23t+C -0.23-t+C y = e =ee -0.23-4 Let, C = K -0.23-t g(t) Ke ... (1) Let, the initial value of dollar be Yo →y(0) = Y Putting t= 0 and y = Yo in (1) we get

$\\\\ Y_0 = K e^{-0.23\cdot 0}= K \\\\ \Rightarrow K = Y_0 \;\;\; \ldots(2)\\\\ $Substituting (2) into (1) we get$\\\\ y(t) = Y_0 e^{-0.23\cdot t}\;\;\;\ldots(3) \\\\ $Let the half life time be $\tau \\\\ \Rightarrow y(\tau) = \frac{y(0)}2 \\\\ \Rightarrow y(\tau)= \frac{Y_0}{2}\\\\ $Substituting $t= \tau $ and $y=\frac{Y_0}{2} $ in (3) we get$\\\\ \frac{Y_0}{2} = Y_0 e^{-0.23\cdot \tau}\\\\ \Rightarrow \frac12 = e^{-0.23\cdot \tau}\\\\ \Rightarrow e^{0.23\cdot \tau} = 2\\\\ \Rightarrow 0.23\cdot \tau = \ln 2\\\\ \Rightarrow \tau = \frac{\ln 2}{0.23} \simeq 3.014 \simeq 3 \;\;\;\;$(Rounded to nearest tenth)$\\\\ \mathbf{The \;required \;answer \;you\; need \;to \;enter \;is:\;\;3 }$