Answer :
To find the number of hours it takes for the bacteria population to reach 3700, we start with the given function:
[tex]\[ P(h) = 2900 e^{0.09h} \][/tex]
We want to determine the value of [tex]\( h \)[/tex] when [tex]\( P(h) \)[/tex] equals 3700. This means we need to solve the equation:
[tex]\[ 2900 e^{0.09h} = 3700 \][/tex]
Here’s how to solve it step-by-step:
1. Divide both sides by 2900:
[tex]\[
e^{0.09h} = \frac{3700}{2900}
\][/tex]
2. Calculate the right-hand side:
[tex]\[
\frac{3700}{2900} \approx 1.2759
\][/tex]
3. Apply the natural logarithm (ln) to both sides to solve for [tex]\( h \)[/tex]:
[tex]\[
\ln(e^{0.09h}) = \ln(1.2759)
\][/tex]
4. Simplify using the property of logarithms:
[tex]\[
0.09h = \ln(1.2759)
\][/tex]
5. Solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{\ln(1.2759)}{0.09}
\][/tex]
6. Calculate the value:
[tex]\[
h \approx 2.7
\][/tex]
Therefore, it will take approximately 2.7 hours for the bacteria population to reach 3700.
[tex]\[ P(h) = 2900 e^{0.09h} \][/tex]
We want to determine the value of [tex]\( h \)[/tex] when [tex]\( P(h) \)[/tex] equals 3700. This means we need to solve the equation:
[tex]\[ 2900 e^{0.09h} = 3700 \][/tex]
Here’s how to solve it step-by-step:
1. Divide both sides by 2900:
[tex]\[
e^{0.09h} = \frac{3700}{2900}
\][/tex]
2. Calculate the right-hand side:
[tex]\[
\frac{3700}{2900} \approx 1.2759
\][/tex]
3. Apply the natural logarithm (ln) to both sides to solve for [tex]\( h \)[/tex]:
[tex]\[
\ln(e^{0.09h}) = \ln(1.2759)
\][/tex]
4. Simplify using the property of logarithms:
[tex]\[
0.09h = \ln(1.2759)
\][/tex]
5. Solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{\ln(1.2759)}{0.09}
\][/tex]
6. Calculate the value:
[tex]\[
h \approx 2.7
\][/tex]
Therefore, it will take approximately 2.7 hours for the bacteria population to reach 3700.
