The number of bacteria \( P(h) \) in a certain population increases according to the following function, where time \( h \) is measured in hours. \[ P(h)=2600 e^{0.06} \text { h } \] 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 out how many hours it will take for the bacteria to reach a population of 3200, we start with the equation: \[ P(h) = 2600 e^{0.06h} \] Setting this equal to 3200 gives us: \[ 3200 = 2600 e^{0.06h} \] Next, we divide both sides by 2600: \[ \frac{3200}{2600} = e^{0.06h} \] This simplifies to: \[ \frac{32}{26} = e^{0.06h} \] Calculating the fraction: \[ \frac{32}{26} = \frac{16}{13} \approx 1.230769 \] Now we take the natural logarithm of both sides: \[ \ln\left(\frac{16}{13}\right) = 0.06h \] Solving for \( h \): \[ h = \frac{\ln\left(\frac{16}{13}\right)}{0.06} \] Calculating \( \ln\left(\frac{16}{13}\right) \): \[ \ln\left(\frac{16}{13}\right) \approx 0.223143 \] Now we substitute this back into our equation for \( h \): \[ h = \frac{0.223143}{0.06} \approx 3.71872 \] Rounding to the nearest tenth, we find: \[ h \approx 3.7 \] So, it will take approximately **3.7 hours** for the number of bacteria to reach 3200.