AM-1555

Exponential and logarithmic models

A Math · Secondary 4 · Hard · Exponential and logarithmic models

The number of bacteria,

N

, in a culture after

t

hours is modelled by

N = p e^{0.4t} + q

, where

p

and

q

are constants. Initially there are 300 bacteria, and after 2 hours there are 900 bacteria.

(a) Find the values of

p

and

q

.

(b) Hence find the number of hours it takes for the bacteria count to first reach 5000.

(a) $p=\dfrac{600}{e^{0.8}-1},\ q=300-\dfrac{600}{e^{0.8}-1}$ ; (b) $t=\dfrac{1}{0.4}\ln\dfrac{5000-q}{p}$ hours
(a) $p=\dfrac{600}{e^{0.4}-1},\ q=300-\dfrac{600}{e^{0.4}-1}$ ; (b) $t=0.4\ln\dfrac{5000-q}{p}$ hours
(a) $p=\dfrac{300}{e^{0.8}-1},\ q=600-\dfrac{300}{e^{0.8}-1}$ ; (b) $t=\dfrac{1}{0.8}\ln\dfrac{5000-q}{p}$ hours
(a) $p=\dfrac{900}{e^{0.8}-1},\ q=300$ ; (b) $t=\dfrac{1}{0.4}\ln\dfrac{5000}{p}$ hours

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