The surface area of a circular pond of radius 'a' is being covered by weeds. The weeds are growing in a circular region whose centre is at the centre of the pond. At time 't' the regoin covered by the weeds has radius 'r' and area 'A'. An ecologist models the growth of the weeds by assuming that the rate of increase of the area covered is proportional to the area of the pond not yet covered.
(i) Show that dA/dt = 2(pi)r dr/dt
(ii) Hence show that the ecologist's model leads to the differential equation, 2r (dr/dt) = k(a^2 - r^2), where k is a constant.
(iii) By solving the differential equation in part (ii), express 'r' in terms of 't', 'a' and 'k', given that r=0 when t=0.
(iv) Will the weeds ever cover the whole pond? Justify your answer.
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Thanks to you for reading this far. I've managed to do parts (i) and (ii) but am stuck on (iii). I've got the markscheme but even after looking at that I don't know what to do. A clear walkthrough would be appreciated.
Answers are (for (iii) and (iv)), r = a sqrt(1 - e^-kt);
No since r->a as t->infinity