Constructing differential equationsEdexcel A-Level Maths: Mind map

Exam questions on Constructing differential equations

1. A population of insects has $N$ individuals at time $t$ days. The rate of increase of $N$ is proportional to $N$. Initially $N=500$ and the population is increasing at $60$ insects per day.
   Find the rate of increase of the population when $N=2000$.2 marks
2. A spherical ball of ice melts so that its radius $r$ cm decreases at a rate that is inversely proportional to the square of the radius. Time $t$ is measured in minutes. When $r=2$, the radius is decreasing at $0.1$ cm per minute.
   Find the rate at which the radius is decreasing when $r=0.5$.2 marks
3. Water flows into a tank at a constant rate of $0.6$ m$^3$ per minute and leaves through a hole in the base at a rate of $0.1\sqrt h$ m$^3$ per minute, where $h$ m is the depth of the water at time $t$ minutes. The tank has a horizontal cross-section of area $2$ m$^2$, so the volume of water in the tank is $V=2h$ m$^3$.
   Show that $\frac{dh}{dt}=0.3-0.05\sqrt h$.3 marks