After 5 hours, the number of bacteria is \( N = 500 \times (1.12)^5 \).

["Title: Understanding Bacterial Growth: After 5 Hours, the Population Reaches ( N = 500 \ imes (1.12)^5 )", "Bacterial growth is a fascinating and critical topic in microbiology, medicine, and environmental science. Understanding how bacteria multiply helps in managing infections, food safety, wastewater treatment, and biotechnology. One common model used to describe bacterial growth in ideal conditions is exponential growth—where the population increases at a constant percentage rate over time.", "### The Exponential Growth Formula", "The formula commonly used to model bacterial growth is:", "[\nN = N_0 \ imes (1 + r)^t\n]", "Where:\n- ( N ) = final population size after time ( t )\n- ( N_0 ) = initial population\n- ( r ) = growth rate per time unit\n- ( t ) = time elapsed", "In our case, the bacterial population after 5 hours is given by the expression:", "[\nN = 500 \ imes (1.12)^5\n]", "This tells us the bacteria started with an initial count of 500, grow at a rate of 12% per hour, and after five hours, their population reaches:", "[\nN = 500 \ imes 1.12^5\n]", "### How Bacterial Growth is Calculated", "Plugging in the values:\n( 1.12^5 \approx 1.7623 ) (computed using exponential powers)", "So,", "[\nN \approx 500 \ imes 1.7623 = 881.15\n]", "Thus, after 5 hours, the bacterial count exceeds 881, demonstrating a more than twofold increase from the initial 500 bacteria.", "### Why 12% Growth Rate?", "A 12% hourly growth rate means the population increases by 12% of the current amount each hour. This exponential behavior is typical in controlled environments with abundant nutrients and no environmental stress—such as in a lab culture or early infection phase.", "### Real-World Implications", "- Medical Field: Rapid bacterial multiplication can lead to severe infections before symptoms develop, emphasizing the need for prompt antibiotic treatment.\n- Food Safety: Understanding growth helps control spoilage microorganisms in refrigerated or shelf-stored foods.\n- Biotechnology: Controlled bacterial growth supports fermentation processes for producing pharmaceuticals or biofuels.", "### Conclusion", "The model ( N = 500 \ imes (1.12)^5 ) elegantly captures the power of exponential bacterial growth. After just 5 hours, a population starting from just 500 bacteria can grow to over 880, emphasizing how quickly microorganisms can multiply under favorable conditions. Grasping these dynamics is essential for science, healthcare, and industry.", "For faster, accurate growth calculations, always apply the precise exponential formula relevant to your time and rate parameters—this model remains foundational in microbiological research.", "---", "Keywords: bacterial growth model, exponential growth formula, ( N = 500 \ imes (1.12)^5 ), microbiology, population increase, bacteria doubling time\n関連分野: 感染症制御、食品衛生、微生物学、バイオテクノロジー"]