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Answer :
To determine how many hours it will take for the number of bacteria to reach 3700, we can use the given function for the population of bacteria:
[tex]\[ P(h) = 2800 e^{0.09 h} \][/tex]
We want to find the value of [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex]. Therefore, we set the equation:
[tex]\[ 2800 e^{0.09 h} = 3700 \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Divide both sides by 2800:
[tex]\[ e^{0.09 h} = \frac{3700}{2800} \][/tex]
2. Simplify the fraction:
[tex]\[ e^{0.09 h} = 1.3214 \][/tex]
3. Take the natural logarithm (ln) of both sides to eliminate the exponential:
[tex]\[ \ln(e^{0.09 h}) = \ln(1.3214) \][/tex]
Using the property of logarithms, [tex]\( \ln(e^x) = x \)[/tex], this simplifies to:
[tex]\[ 0.09 h = \ln(1.3214) \][/tex]
4. Solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{\ln(1.3214)}{0.09} \][/tex]
5. Calculate the value of [tex]\( h \)[/tex]:
Evaluate the natural logarithm and division:
[tex]\[ h \approx 3.1 \][/tex]
Therefore, it will take approximately 3.1 hours for the number of bacteria to reach 3700.
[tex]\[ P(h) = 2800 e^{0.09 h} \][/tex]
We want to find the value of [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex]. Therefore, we set the equation:
[tex]\[ 2800 e^{0.09 h} = 3700 \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Divide both sides by 2800:
[tex]\[ e^{0.09 h} = \frac{3700}{2800} \][/tex]
2. Simplify the fraction:
[tex]\[ e^{0.09 h} = 1.3214 \][/tex]
3. Take the natural logarithm (ln) of both sides to eliminate the exponential:
[tex]\[ \ln(e^{0.09 h}) = \ln(1.3214) \][/tex]
Using the property of logarithms, [tex]\( \ln(e^x) = x \)[/tex], this simplifies to:
[tex]\[ 0.09 h = \ln(1.3214) \][/tex]
4. Solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{\ln(1.3214)}{0.09} \][/tex]
5. Calculate the value of [tex]\( h \)[/tex]:
Evaluate the natural logarithm and division:
[tex]\[ h \approx 3.1 \][/tex]
Therefore, it will take approximately 3.1 hours for the number of bacteria to reach 3700.
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