Dr. Vance models the growth of a deep-sea extremophile using the function \( N(t) = 500 \cdot e^{0.3t} \), where \( t \) is time in hours. How many bacteria are present after 6 hours? Round to the nearest whole number.
["Dr. Vance Models Deep-Sea Extremophile Growth with Exponential Function", "In the mysterious depths of the ocean, where crushing pressures and darkness define the environment, deep-sea extremophiles thrive in conditions once thought uninhabitable. Dr. Vance, a leading marine microbiologist, uses exponential growth modeling to predict how these remarkable bacteria multiply over time. His work provides critical insights into life’s resilience and potential for biotechnological breakthroughs.", "### The Growth Function: ( N(t) = 500 \cdot e^{0.3t} )", "Dr. Vance models bacterial population growth using the function\n[ N(t) = 500 \cdot e^{0.3t} ]\nwhere:\n- ( N(t) ) represents the number of bacteria at time ( t ) (in hours),\n- ( t = 0 ) corresponds to the initial moment,\n- ( 500 ) is the starting population,\n- ( 0.3 ) is the growth rate constant, reflecting rapid doubling under extreme conditions.", "This model captures how extremophiles sustain exponential increases in nutrient-rich deep-sea vents, doubling so quickly that population surges in just hours.", "### Calculating Population After 6 Hours", "To determine how many bacteria are present after 6 hours, substitute ( t = 6 ) into the model:\n[ N(6) = 500 \cdot e^{0.3 \ imes 6} = 500 \cdot e^{1.8} ]", "Using a calculator,\n[ e^{1.8} \approx 6.0496 ]", "Now compute:\n[ N(6) \approx 500 \ imes 6.0496 = 3024.8 ]", "Rounding to the nearest whole number:\n[ N(6) \approx 3025 ]", "### Conclusion", "After 6 hours, the deep-sea extremophile population reaches approximately 3,025 bacteria—a testament to life’s adaptability in Earth’s most extreme habitats. Dr. Vance’s model not only quantifies this growth but also supports broader ecological and astrobiological research into survival strategies beyond surface conditions.", "Understanding such microbial dynamics may one day inform experiments in space habitats or bioreactor design, where controlled extremophile growth holds transformative promise.", "---", "Keywords: deep-sea extremophiles, bacterial growth model, exponential growth, Dr. Vance, ( N(t) = 500 \cdot e^{0.3t} ), ocean microbiology, population dynamics, scientific modeling, exponential function, marine science."]
