To determine how many hours it will take for the number of bacteria to reach 3000, we will start with the given equation that represents the population of bacteria as a function of time [tex]\( h \)[/tex] in hours:
[tex]\[ P(h) = 2200 e^{0.11 h} \][/tex]
We need to find the time [tex]\( h \)[/tex] when [tex]\( P(h) = 3000 \)[/tex]. So we set up the equation:
[tex]\[ 3000 = 2200 e^{0.11 h} \][/tex]
Next, we isolate the exponential term by dividing both sides of the equation by 2200:
[tex]\[ \frac{3000}{2200} = e^{0.11 h} \][/tex]
Simplifying the fraction gives us:
[tex]\[ \frac{3000}{2200} = \frac{15}{11} \approx 1.363636 \][/tex]
So the equation is now:
[tex]\[ 1.363636 = e^{0.11 h} \][/tex]
To solve for [tex]\( h \)[/tex], we take the natural logarithm (ln) of both sides:
[tex]\[ \ln(1.363636) = \ln(e^{0.11 h}) \][/tex]
Using the property of logarithms [tex]\( \ln(e^x) = x \)[/tex], we get:
[tex]\[ \ln(1.363636) = 0.11 h \][/tex]
Now, we solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{\ln(1.363636)}{0.11} \][/tex]
Evaluating the natural logarithm and the division:
[tex]\[ \ln(1.363636) \approx 0.310154 \][/tex]
[tex]\[ h = \frac{0.310154}{0.11} \approx 2.819590 \][/tex]
Therefore, the time [tex]\( h \)[/tex] required for the number of bacteria to reach 3000 is approximately:
[tex]\[ h \approx 2.819590 \][/tex]
Rounding to the nearest tenth:
[tex]\[ h \approx 2.8 \][/tex]
So, it will take approximately:
[tex]\[ \boxed{2.8} \][/tex]
hours for the number of bacteria to reach 3000.
