Questions: The number of bacteria P(h) in a certain population increases according to the following function, where time h is measured in hours. P(h) = 2100 e^(0.16 h) How many hours will it take for the number of bacteria to reach 3000? Round your answer to the nearest tenth, and do not round any intermediate computations. (1) hours

Solution Steps

To find the number of hours it will take for the number of bacteria to reach 3000, we need to solve the equation \( P(h) = 3000 \) for \( h \). Given the function \( P(h) = 2100 e^{0.16h} \), we set it equal to 3000 and solve for \( h \) using logarithms. This involves isolating the exponential term and then taking the natural logarithm of both sides to solve for \( h \).

We start with the function that describes the population of bacteria: \[ P(h) = 2100 e^{0.16h} \] To find the time \( h \) when the population reaches 3000, we set up the equation: \[ 2100 e^{0.16h} = 3000 \]

Next, we isolate the exponential term by dividing both sides by 2100: \[ e^{0.16h} = \frac{3000}{2100} \] Calculating the right side gives: \[ e^{0.16h} = \frac{3000}{2100} \approx 1.4285714285714286 \]

We take the natural logarithm of both sides to solve for \( h \): \[ 0.16h = \ln\left(1.4285714285714286\right) \] Calculating the natural logarithm: \[ 0.16h \approx 0.35667494394 \]

Now, we solve for \( h \) by dividing both sides by 0.16: \[ h \approx \frac{0.35667494394}{0.16} \approx 2.2292183996170776 \] Rounding this value to the nearest tenth gives: \[ h \approx 2.2 \]

Final Answer