Answer :
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.
[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.