The number of bacteria [tex] P(h) [/tex] in a certain population increases according to the following function, where time [tex] h [/tex] is measured in hours:

\[ P(h) = 2800 e^{0.09 h} \]

How many hours will it take for the number of bacteria to reach 3700? Round your answer to the nearest tenth, and do not round any intermediate computations.

[tex]$\square$[/tex] hours

To solve the problem of determining how many hours it will take for the number of bacteria to reach 3700, we use the given formula for the bacterial population:

[tex]\[ P(h) = 2800 \cdot e^{0.09h} \][/tex]

To find the time [tex]$ h $[/tex] when [tex]$ P(h) = 3700 $[/tex], we need to solve the equation:

[tex]\[ 3700 = 2800 \cdot e^{0.09h} \][/tex]

Step-by-step solution:

1. Divide both sides by 2800 to isolate the exponential term:
[tex]\[
\frac{3700}{2800} = e^{0.09h}
\][/tex]

2. Simplify the left-hand side:
[tex]\[
1.32142857 \approx e^{0.09h}
\][/tex]

3. Take the natural logarithm (ln) of both sides to solve for [tex]$ h $[/tex]:
[tex]\[
\ln(1.32142857) = \ln(e^{0.09h})
\][/tex]

4. Apply the property of logarithms where [tex]$\ln(e^x) = x$[/tex]:
[tex]\[
\ln(1.32142857) = 0.09h
\][/tex]

5. Solve for [tex]$ h $[/tex] by dividing both sides by 0.09:
[tex]\[
h = \frac{\ln(1.32142857)}{0.09}
\][/tex]

6. Calculate the natural logarithm and perform the division:
- Compute [tex]$\ln(1.32142857) \approx 0.2784$[/tex] (using a calculator for the precise value)
- Then divide by [tex]$ 0.09 $[/tex]:
[tex]\[
h \approx \frac{0.2784}{0.09} \approx 3.09681558298912
\][/tex]

7. Round the result to the nearest tenth:
[tex]\[
h \approx 3.1
\][/tex]

Therefore, it will take approximately 3.1 hours for the number of bacteria to reach 3700.