Finding the time given an exponential function with base e that... The number of bacteria \( P(h) \) in a certain population increases according to the following function, where time \( h \) is measured in hours. \[ P(h)=1700 e^{0.19} h \] How many hours will it take for the number of bacteria to reach 2400 ? Round your answer to the nearest tenth, and do not round any intermediate computations. \( \square \) hours

To find out how long it takes for the bacteria population to reach 2400, you want to set up the equation: \[ 2400 = 1700 e^{0.19h}. \] First, divide both sides by 1700: \[ \frac{2400}{1700} = e^{0.19h}. \] This simplifies to: \[ \frac{24}{17} = e^{0.19h}. \] Now, take the natural logarithm of both sides to solve for \( h \): \[ \ln\left(\frac{24}{17}\right) = 0.19h. \] Next, isolate \( h \): \[ h = \frac{\ln\left(\frac{24}{17}\right)}{0.19}. \] Calculating the values gives: \[ h \approx \frac{0.1744}{0.19} \approx 0.918. \] Finally, rounding to the nearest tenth results in: \( \approx 0.9 \) hours. So, in a little under an hour, your bacteria are set to multiply! It’s fascinating how fast microorganisms can grow!