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.
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Beyond the Answer
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.
