Question

This exercise uses the population growth model. It is observed that a certain bacteria culture has a relative growth rate of 15% per hour, but in the presence of an antibiotic the relative growth rate is reduced to 8% per hour. The initial number of bacteria in the culture is 28. Find the projected population after 24 hours for the following conditions. (Round your answers to the nearest whole number.) (a) No antibiotic is present, so the relative growth rate is 15%. (b) An antibiotic is present in the culture, so the relative growth rate is reduced to 8%.

Answer 1

a) P(24) = 1025.

b) P(24) = 191.

Explanation:

This population can be modeled by the following exponential model.

`P(t) = P_(0)e^(rt)`

In which P(t) is the population after t hours,
`P_(0)` is the initial population and r is the decimal growth rate.

The initial number of bacteria in the culture is 28. This means that
`P_(0) = 28`.

Population after 24 hours.

(a) No antibiotic is present, so the relative growth rate is 15%.

So r = 0.15.

`P(t) = P_(0)e^(rt)`

`P(24) = 28e^(0.15*24) = 1024.75`

(b) An antibiotic is present in the culture, so the relative growth rate is reduced to 8%.

So r = 0.08.

`P(t) = P_(0)e^(rt)`

`P(24) = 28e^(0.08*24) = 190.99`