Answer :
To find out how many hours it will take for the number of bacteria to reach 3700, we'll use the given exponential function:
[tex]\[ P(h) = 2800 \cdot e^{0.08h} \][/tex]
where [tex]\( P(h) \)[/tex] is the number of bacteria after [tex]\( h \)[/tex] hours, and [tex]\( e \)[/tex] is the base of the natural logarithm. We need to find the time [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex].
Here are the steps to solve this problem:
1. Set the Equation: Start by setting the function equal to the target population:
[tex]\[ 3700 = 2800 \cdot e^{0.08h} \][/tex]
2. Solve for the Exponential Part: Divide both sides by 2800 to isolate the exponential expression:
[tex]\[ e^{0.08h} = \frac{3700}{2800} \][/tex]
3. Calculate the Quotient: Simplify the fraction:
[tex]\[ e^{0.08h} = 1.3214 \][/tex]
4. Use Logarithms to Find [tex]\( h \)[/tex]: Take the natural logarithm (ln) on both sides of the equation to solve for [tex]\( h \)[/tex]:
[tex]\[ \ln(e^{0.08h}) = \ln(1.3214) \][/tex]
Since [tex]\(\ln(e^x) = x\)[/tex], this simplifies to:
[tex]\[ 0.08h = \ln(1.3214) \][/tex]
5. Calculate [tex]\( h \)[/tex]: Divide by 0.08 to isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{\ln(1.3214)}{0.08} \][/tex]
6. Compute the Numerical Value: The value of [tex]\(\ln(1.3214)\)[/tex] is approximately 0.2782, so:
[tex]\[ h \approx \frac{0.2782}{0.08} \approx 3.48275 \][/tex]
7. Round to the Nearest Tenth: Finally, rounding 3.48275 to the nearest tenth gives you:
[tex]\[ h \approx 3.5 \][/tex]
Therefore, it will take approximately 3.5 hours for the number of bacteria to reach 3700.
[tex]\[ P(h) = 2800 \cdot e^{0.08h} \][/tex]
where [tex]\( P(h) \)[/tex] is the number of bacteria after [tex]\( h \)[/tex] hours, and [tex]\( e \)[/tex] is the base of the natural logarithm. We need to find the time [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex].
Here are the steps to solve this problem:
1. Set the Equation: Start by setting the function equal to the target population:
[tex]\[ 3700 = 2800 \cdot e^{0.08h} \][/tex]
2. Solve for the Exponential Part: Divide both sides by 2800 to isolate the exponential expression:
[tex]\[ e^{0.08h} = \frac{3700}{2800} \][/tex]
3. Calculate the Quotient: Simplify the fraction:
[tex]\[ e^{0.08h} = 1.3214 \][/tex]
4. Use Logarithms to Find [tex]\( h \)[/tex]: Take the natural logarithm (ln) on both sides of the equation to solve for [tex]\( h \)[/tex]:
[tex]\[ \ln(e^{0.08h}) = \ln(1.3214) \][/tex]
Since [tex]\(\ln(e^x) = x\)[/tex], this simplifies to:
[tex]\[ 0.08h = \ln(1.3214) \][/tex]
5. Calculate [tex]\( h \)[/tex]: Divide by 0.08 to isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{\ln(1.3214)}{0.08} \][/tex]
6. Compute the Numerical Value: The value of [tex]\(\ln(1.3214)\)[/tex] is approximately 0.2782, so:
[tex]\[ h \approx \frac{0.2782}{0.08} \approx 3.48275 \][/tex]
7. Round to the Nearest Tenth: Finally, rounding 3.48275 to the nearest tenth gives you:
[tex]\[ h \approx 3.5 \][/tex]
Therefore, it will take approximately 3.5 hours for the number of bacteria to reach 3700.
