Mathematics - Application of Derivative Question with Solution | TestHub

Given differential equation is $\frac{dp(t)}{dt}=0.5p(t)-450$ $\Rightarrow \frac{dp(t)}{dt}=\frac{1}{2}p(t)-450$ $\Rightarrow \frac{dp(t)}{dt}=\frac{p(t)-900}{2}$ $\Rightarrow 2\frac{dp(t)}{dt}=-[900-p(t)]$ $\Rightarrow 2\frac{dp(t)}{900-p(t)}=-dt$ Integrate both the side, we get : $-2\int \frac{dp(t)}{900-p(t)}=\int dt$ Let $900-p(t)=u\Rightarrow -dp(t)=du$ $\therefore$ We have; $2\int \frac{du}{u}=\int dt\Rightarrow 2\ln u=t+c$ $\Rightarrow 2\ln [900-p(t)]=t+c$ when $t=0,p(0)=850$ $2\ln (50)=c\Rightarrow 2\left[\ln \left(\frac{900-p(t)}{50}\right)\right]=t$ $\Rightarrow 900-p(t)=50e^{\frac{t}{2}}$ $\Rightarrow p(t)=900-50e^{\frac{t}{2}}$ let $p\left(t_1\right)=0$ $0=900-50e^{\frac{t_1}{2}}\therefore t_1=2\ln 18$