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Answer :
To find out how many hours it will take for the bacteria population to reach 3,700, we can use the given exponential growth function:
[tex]\[ P(h) = 2800 \cdot e^{0.08h} \][/tex]
We are trying to find the time [tex]\( h \)[/tex] when the population [tex]\( P(h) \)[/tex] is 3,700. So, we set up the equation:
[tex]\[ 3700 = 2800 \cdot e^{0.08h} \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Isolate the exponential component: Divide both sides of the equation by 2800 to isolate the exponential expression.
[tex]\[ \frac{3700}{2800} = e^{0.08h} \][/tex]
2. Simplify the fraction: Simplify [tex]\( \frac{3700}{2800} \)[/tex].
[tex]\[ 1.321428571 \approx e^{0.08h} \][/tex]
3. Apply the natural logarithm: To solve for [tex]\( h \)[/tex], take the natural logarithm of both sides. Remember [tex]\( \ln(e^x) = x \)[/tex].
[tex]\[ \ln(1.321428571) = \ln(e^{0.08h}) \][/tex]
4. Simplify the equation using properties of logarithms:
[tex]\[ \ln(1.321428571) = 0.08h \][/tex]
5. Solve for [tex]\( h \)[/tex]: Divide both sides by 0.08 to solve for [tex]\( h \)[/tex].
[tex]\[ h = \frac{\ln(1.321428571)}{0.08} \][/tex]
6. Calculate the value: Compute the value using a calculator to find [tex]\( h \)[/tex].
[tex]\[ h \approx 3.4839175308627564 \][/tex]
7. Round the result: Finally, round the result to the nearest tenth.
[tex]\[ h \approx 3.5 \][/tex]
So, it will take approximately 3.5 hours for the bacteria population to reach 3,700.
[tex]\[ P(h) = 2800 \cdot e^{0.08h} \][/tex]
We are trying to find the time [tex]\( h \)[/tex] when the population [tex]\( P(h) \)[/tex] is 3,700. So, we set up the equation:
[tex]\[ 3700 = 2800 \cdot e^{0.08h} \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Isolate the exponential component: Divide both sides of the equation by 2800 to isolate the exponential expression.
[tex]\[ \frac{3700}{2800} = e^{0.08h} \][/tex]
2. Simplify the fraction: Simplify [tex]\( \frac{3700}{2800} \)[/tex].
[tex]\[ 1.321428571 \approx e^{0.08h} \][/tex]
3. Apply the natural logarithm: To solve for [tex]\( h \)[/tex], take the natural logarithm of both sides. Remember [tex]\( \ln(e^x) = x \)[/tex].
[tex]\[ \ln(1.321428571) = \ln(e^{0.08h}) \][/tex]
4. Simplify the equation using properties of logarithms:
[tex]\[ \ln(1.321428571) = 0.08h \][/tex]
5. Solve for [tex]\( h \)[/tex]: Divide both sides by 0.08 to solve for [tex]\( h \)[/tex].
[tex]\[ h = \frac{\ln(1.321428571)}{0.08} \][/tex]
6. Calculate the value: Compute the value using a calculator to find [tex]\( h \)[/tex].
[tex]\[ h \approx 3.4839175308627564 \][/tex]
7. Round the result: Finally, round the result to the nearest tenth.
[tex]\[ h \approx 3.5 \][/tex]
So, it will take approximately 3.5 hours for the bacteria population to reach 3,700.
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