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 the time it will take for the number of bacteria to reach 2400, we need to solve the equation \( P(h) = 2400 \) for \( h \). Given the exponential function: \[ P(h) = 1700 e^{0.19h} \] We want to find \( h \) when \( P(h) = 2400 \). Substitute \( P(h) = 2400 \) into the equation: \[ 2400 = 1700 e^{0.19h} \] Now, we can solve for \( h \) using the equation above. Solve the equation by following steps: - step0: Solve for \(h\): \(2400=1700e^{0.19h}\) - step1: Convert the expressions: \(2400=1700e^{\frac{19}{100}h}\) - step2: Swap the sides: \(1700e^{\frac{19}{100}h}=2400\) - step3: Divide both sides: \(\frac{1700e^{\frac{19}{100}h}}{1700}=\frac{2400}{1700}\) - step4: Divide the numbers: \(e^{\frac{19}{100}h}=\frac{24}{17}\) - step5: Take the logarithm of both sides: \(\ln{\left(e^{\frac{19}{100}h}\right)}=\ln{\left(\frac{24}{17}\right)}\) - step6: Evaluate the logarithm: \(\frac{19}{100}h=\ln{\left(\frac{24}{17}\right)}\) - step7: Multiply by the reciprocal: \(\frac{19}{100}h\times \frac{100}{19}=\ln{\left(\frac{24}{17}\right)}\times \frac{100}{19}\) - step8: Multiply: \(h=\frac{100\ln{\left(\frac{24}{17}\right)}}{19}\) - step9: Simplify: \(h=\frac{100\ln{\left(24\right)}-100\ln{\left(17\right)}}{19}\) The solution to the equation \( 2400 = 1700 e^{0.19h} \) is: \[ h = \frac{100\ln{(24)}-100\ln{(17)}}{19} \] Now, we can calculate the value of \( h \) by substituting the values into the equation. Calculate or simplify the expression \( 19 \) Numbers by following steps: - step0: Solution: \(19\) Calculate or simplify the expression \( 34.484049/19 \) Calculate the value by following steps: - step0: Calculate: \(\frac{34.484049}{19}\) - step1: Convert the expressions: \(\frac{\frac{34484049}{1000000}}{19}\) - step2: Multiply by the reciprocal: \(\frac{34484049}{1000000}\times \frac{1}{19}\) - step3: Multiply the fractions: \(\frac{34484049}{1000000\times 19}\) - step4: Multiply: \(\frac{34484049}{19000000}\) The time it will take for the number of bacteria to reach 2400 is approximately 1.815 hours when rounded to the nearest tenth.