5.14 Differential equations: separation of variablesIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.14 Differential equations: separation of variables

1. The mass $G$ grams of algae in a pond at time $t$ days grows at a rate proportional to $G$, so $\frac{dG}{dt}=kG$ where $k>0$ is a constant. Initially $G=50$, and when $t=4$, $G=80$.
   Find the mass of algae after $10$ days, using the unrounded value of $k$.2 marks
2. A sample of a radioactive substance has mass $m$ mg at time $t$ years. The mass decreases at a rate proportional to $m$. Initially $m=200$, and after $5$ years $m=150$.
   Find the value of $k$.2 marks
3. A boat's engine is switched off at $t=0$. For $t\ge0$ the speed $v$ m s$^{-1}$ of the boat satisfies $\frac{dv}{dt}=-0.05v^2$, with $v=10$ when $t=0$.
   Show that $v=\frac{20}{t+2}$.3 marks