Sure, let's work through each part of this question step by step.
### (a) Finding the relative rate of growth, [tex]\( k \)[/tex]
The general form of the exponential growth equation is:
[tex]\[ P(t) = P_0 e^{kt} \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( k \)[/tex] is the relative rate of growth,
- [tex]\( t \)[/tex] is the time,
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex].
We know:
- [tex]\( P_0 = 120 \)[/tex] (initial population),
- [tex]\( P(5) = 600 \)[/tex] (population after 5 hours).
Plugging in these values, we get:
[tex]\[ 600 = 120 e^{5k} \][/tex]
To solve for [tex]\( k \)[/tex]:
[tex]\[ \frac{600}{120} = e^{5k} \][/tex]
[tex]\[ 5 = e^{5k} \][/tex]
[tex]\[ \ln(5) = 5k \][/tex]
[tex]\[ k = \frac{\ln(5)}{5} \approx 0.32189 \][/tex]
So, the value of [tex]\( k \)[/tex] rounded to five decimal places is:
[tex]\[ k \approx 0.32189 \][/tex]
### (b) Expressing the Population [tex]\( P \)[/tex] After [tex]\( t \)[/tex] Hours
Using the value of [tex]\( k \)[/tex] we found in part (a), the function [tex]\( P(t) \)[/tex] will be:
[tex]\[ P(t) = 120 e^{0.32189 t} \][/tex]
### (c) Population After 8 Hours
To find the population after 8 hours, we use the function [tex]\( P(t) \)[/tex]:
[tex]\[ P(8) = 120 e^{0.32189 \times 8} \][/tex]
[tex]\[ P(8) \approx 1575.92 \][/tex]
So, the population after 8 hours will be approximately 1575.92 bacteria.
### (d) Time for Population to Reach 1550
We want to find the time [tex]\( t \)[/tex] when the population will reach 1550. Using the equation:
[tex]\[ 1550 = 120 e^{0.32189 t} \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ \frac{1550}{120} = e^{0.32189 t} \][/tex]
[tex]\[ \ln\left(\frac{1550}{120}\right) = 0.32189 t \][/tex]
[tex]\[ t = \frac{\ln\left(\frac{1550}{120}\right)}{0.32189} \approx 7.95 \][/tex]
So, it will take approximately 7.95 hours for the population to reach 1550 bacteria.
### Summary of Results
(a) [tex]\( k \approx 0.32189 \)[/tex]
(b) [tex]\( P(t) = 120 e^{0.32189 t} \)[/tex]
(c) Population after 8 hours: ~1575.92 bacteria
(d) Time to reach a population of 1550: ~7.95 hours
