Answer :
Let's solve the problem step by step.
### (a) Formula Relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex]
We know that the area covered by a population of bacteria increases according to a continuous exponential growth model. The initial area is [tex]\( 98.1 \, \text{mm}^2 \)[/tex], and the observed doubling time is 13 hours.
The general formula for exponential growth is:
[tex]\[ y(t) = y_0 \cdot e^{rt} \][/tex]
where:
- [tex]\( y(t) \)[/tex] is the area at time [tex]\( t \)[/tex].
- [tex]\( y_0 \)[/tex] is the initial area ([tex]\( 98.1 \, \text{mm}^2 \)[/tex]).
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in hours.
Since the population doubles in the doubling time, we can use the relationship:
[tex]\[ 2 = e^{r \cdot \text{doubling time}} \][/tex]
To find [tex]\( r \)[/tex], take the natural logarithm on both sides:
[tex]\[ \ln(2) = r \cdot \text{doubling time} \][/tex]
Thus, the growth rate [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{\ln(2)}{13} \][/tex]
So, the formula relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex] is:
[tex]\[ y(t) = 98.1 \cdot e^{\left(\frac{\ln(2)}{13}\right)t} \][/tex]
### (b) Area After 8 Hours
Now, we want to find the area of the sample after 8 hours.
Using the formula:
[tex]\[ y(8) = 98.1 \cdot e^{\left(\frac{\ln(2)}{13}\right) \cdot 8} \][/tex]
The exponential growth calculation gives us the area after 8 hours. According to the correct mathematical computation, the area is approximately:
[tex]\[ 150.3 \, \text{mm}^2 \][/tex]
Rounded to the nearest tenth, the area of the sample after 8 hours is [tex]\( 150.3 \, \text{mm}^2 \)[/tex].
### (a) Formula Relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex]
We know that the area covered by a population of bacteria increases according to a continuous exponential growth model. The initial area is [tex]\( 98.1 \, \text{mm}^2 \)[/tex], and the observed doubling time is 13 hours.
The general formula for exponential growth is:
[tex]\[ y(t) = y_0 \cdot e^{rt} \][/tex]
where:
- [tex]\( y(t) \)[/tex] is the area at time [tex]\( t \)[/tex].
- [tex]\( y_0 \)[/tex] is the initial area ([tex]\( 98.1 \, \text{mm}^2 \)[/tex]).
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in hours.
Since the population doubles in the doubling time, we can use the relationship:
[tex]\[ 2 = e^{r \cdot \text{doubling time}} \][/tex]
To find [tex]\( r \)[/tex], take the natural logarithm on both sides:
[tex]\[ \ln(2) = r \cdot \text{doubling time} \][/tex]
Thus, the growth rate [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{\ln(2)}{13} \][/tex]
So, the formula relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex] is:
[tex]\[ y(t) = 98.1 \cdot e^{\left(\frac{\ln(2)}{13}\right)t} \][/tex]
### (b) Area After 8 Hours
Now, we want to find the area of the sample after 8 hours.
Using the formula:
[tex]\[ y(8) = 98.1 \cdot e^{\left(\frac{\ln(2)}{13}\right) \cdot 8} \][/tex]
The exponential growth calculation gives us the area after 8 hours. According to the correct mathematical computation, the area is approximately:
[tex]\[ 150.3 \, \text{mm}^2 \][/tex]
Rounded to the nearest tenth, the area of the sample after 8 hours is [tex]\( 150.3 \, \text{mm}^2 \)[/tex].
