To evaluate the exponential growth formula [tex]\( y = A e^{rn} \)[/tex] with the given values, follow these steps:
1. Identify the given values:
- Initial amount of bacteria, [tex]\( A = 1,610,000 \)[/tex]
- Growth rate per hour, [tex]\( r = 7.1\% \)[/tex] (which can be written as a decimal, [tex]\( r = 0.071 \)[/tex])
- Number of hours, [tex]\( n = 27 \)[/tex]
2. Insert the values into the exponential growth formula:
[tex]\[
y = 1,610,000 \times e^{0.071 \times 27}
\][/tex]
3. Calculate the exponent:
[tex]\[
0.071 \times 27 = 1.917
\][/tex]
Hence the formula becomes:
[tex]\[
y = 1,610,000 \times e^{1.917}
\][/tex]
4. Calculate [tex]\( e^{1.917} \)[/tex]:
The value of [tex]\( e^{1.917} \approx 6.798 \)[/tex] (this is typically obtained using a scientific calculator or a mathematical software).
5. Multiply by the initial amount of bacteria:
[tex]\[
y = 1,610,000 \times 6.798
\][/tex]
[tex]\[
y \approx 10,948,847.27
\][/tex]
6. Round the result to the nearest integer:
[tex]\[
y \approx 10,948,847
\][/tex]
Therefore, the final amount of bacteria after 27 hours is approximately [tex]\( \boxed{10,948,847} \)[/tex].
