Question 8 - A-Level Maths

A pond is initially empty and is then filled gradually with water. After $t$ minutes, the depth of the water, $x$ metres, satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \sqrt { 4 + 5 x } } { 5 ( 1 + t ) ^ { 2 } }$$ Solve this differential equation to find $x$ in terms of $t$.
Another pond is gradually filling with water. After $t$ minutes, the surface of the water forms a circle of radius $r$ metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
1. Write down a differential equation, in the variables $r$ and $t$ and a constant of proportionality, which represents how the radius of the surface of the water is changing with time.
  (You are not required to solve your differential equation.)
2. When the radius of the pond is 1 metre, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.
  [0pt] [2 marks]
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