The number of bacteria \( P(h) \) in a certain population increases according to the following function, where time \( h \) is measured in hours. \[ P(h)=2000 e^{0.06 h} \] How many hours will it take for the number of bacteria to reach 3200 ? Round your answer to the nearest tenth, and do not round any intermediate computations.

To find the number of hours \( h \) it takes for the bacterial population \( P(h) \) to reach 3200, we start with the equation: \[ 2000 e^{0.06 h} = 3200 \] To isolate \( e^{0.06 h} \), we divide both sides by 2000: \[ e^{0.06 h} = \frac{3200}{2000} = 1.6 \] Next, we take the natural logarithm of both sides: \[ \ln(e^{0.06 h}) = \ln(1.6) \] Using the property of logarithms that \( \ln(e^x) = x \), we get: \[ 0.06 h = \ln(1.6) \] Now we can solve for \( h \) by dividing both sides by 0.06: \[ h = \frac{\ln(1.6)}{0.06} \] Calculating \( \ln(1.6) \): \[ \ln(1.6) \approx 0.4700 \] Now substituting back into the equation for \( h \): \[ h \approx \frac{0.4700}{0.06} \approx 7.8333 \] Rounding to the nearest tenth, we find: \[ h \approx 7.8 \] Thus, it will take approximately 7.8 hours for the number of bacteria to reach 3200.