Question

3. The population of bacteria in a culture decreases at a rate proportional to the number of bacteria present at any time t. The initial population is 500 and the population decreases 10% in 1 hour. Determine the half-life of the population of bacteria. How long does it take for the population to be 10? (8 marks ).

Answer 1

Solve. Let Pits be the population of bacteria at any time at hours). since the population decreases at a rate proportional to the number of bacteria present at any time t, therefore a PIN op de k Plt) – us where K is a broportionality constant. is can be rewrite as- Equation .= k dt Plt On integration, we get ln (Plt) = kt+C, or | PH = cekti (let e" - c

Since intially the population is 500, therefore Plo)= 500 and by this 500 - C Also after I hours , the population decreases 10% lie bopulation after 1 hour = 99 x 500 = 450 100 Thus P(1) = 450 and by equation (ii), le get 450 = 500 e >> K= ln(2) • Inlal l e a This equation (ii) becomes . PIU = 500 € (In (7)) + * = 500 (21+ ... [ Plts = 500 (18+)

Let after t hours, the population becomes half of its initial population le Plt) = 500 = 250 * 250 = 500 lglt DŹ - 1994 1. ln ( 2 ) = t ln (2) = t. tn (°15) In 10,5) ln(0.g) - 6.579 O Hence half life time is 16.5 79 hours. het after t hours population be to, then 10 = 500 (13) s 5o = (49+ A te In (1650) ln (0.02) . In 19/10 - Inloop= 37.130 hours.
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