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 \( h \) it takes for the number of bacteria \( P(h) \) to reach 2400, we start with the given exponential function: \[ P(h) = 1700 e^{0.19 h} \] We need to set \( P(h) \) equal to 2400 and solve for \( h \): \[ 1700 e^{0.19 h} = 2400 \] ### Step 1: Isolate the exponential term First, we divide both sides by 1700: \[ e^{0.19 h} = \frac{2400}{1700} \] Calculating the right side: \[ \frac{2400}{1700} = \frac{24}{17} \approx 1.411764705882353 \] ### Step 2: Take the natural logarithm of both sides Next, we take the natural logarithm (ln) of both sides: \[ \ln(e^{0.19 h}) = \ln\left(\frac{24}{17}\right) \] Using the property of logarithms that \( \ln(e^x) = x \): \[ 0.19 h = \ln\left(\frac{24}{17}\right) \] ### Step 3: Solve for \( h \) Now, we can solve for \( h \): \[ h = \frac{\ln\left(\frac{24}{17}\right)}{0.19} \] ### Step 4: Calculate the value of \( h \) Now, we will calculate \( h \) using the natural logarithm: \[ h = \frac{\ln\left(\frac{24}{17}\right)}{0.19} \] Let's compute this value. Calculate the value by following steps: - step0: Calculate: \(\frac{\ln{\left(\frac{24}{17}\right)}}{0.19}\) - step1: Rewrite the expression: \(\frac{\ln{\left(\frac{24}{17}\right)}}{\frac{19}{100}}\) - step2: Multiply by the reciprocal: \(\ln{\left(\frac{24}{17}\right)}\times \frac{100}{19}\) - step3: Multiply the terms: \(\frac{\ln{\left(\frac{24}{17}\right)}\times 100}{19}\) - step4: Multiply the terms: \(\frac{100\ln{\left(\frac{24}{17}\right)}}{19}\) Round the number \( 1.81495 \) to nearest tenth. Round the number by following steps: - step0: Round \(1.81495\) to \(1\) decimal places using standard rounding method: \(1.81495\) - step1: Round \(1.81495\) to \(1\) decimal places using standard rounding method\(:\) \(1.8\) After calculating, we find that: \[ h \approx 1.81495 \] Rounding this to the nearest tenth, we get: \[ h \approx 1.8 \] Thus, it will take approximately \( \boxed{1.8} \) hours for the number of bacteria to reach 2400.