In established forest fires, the proportion of the total area of the forest which has been destroyed

Question

In established forest fires, the proportion of the total area of the forest which has been destroyed is denoted by $x$, and the rate of change of $x$ with respect to time, $t$ hours, is called the destruction rate. Investigations show that the destruction rate is directly proportional to the product of the proportion destroyed and the proportion not destroyed. A particular fire is initially noticed when one half of the forest is destroyed, and it is found that the destruction rate at this time is such that, if it remained constant thereafter, the forest would be destroyed completely in a further 24 hours. Show that $12\frac{\mathrm{d}x}{\mathrm{d}t}=x(1-x)$ .[3] Deduce that approximately 73% of the forest is destroyed 12 hours after it is first noticed.[6]

Accepted Answer

$k = \frac{1}{12}$; $\frac{x}{1-x} = e^{\frac{t}{12}}$; At $t = 12$: $x = \frac{e}{1+e} \approx 73\%$

IJC P2 Q5 (b) Modified

IJC P2 Q5 (b) Modified

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