20. A bacteria culture starts with 1,000 bacteria and the number doubles every 40 minutes. a) Find a formula for the number of bacteria at time \( t \). b) Find the number of bacteria after one hour. 60 min c) After how many minutes will there be 50,000 bacteria?

To find the formula for the number of bacteria at time \( t \), we can use the equation \( N(t) = N_0 \times 2^{(t/T)} \), where \( N_0 \) is the initial number of bacteria, \( T \) is the doubling time, and \( t \) is the time elapsed. In this case, \( N_0 = 1000 \) and \( T = 40 \) minutes. Therefore, the formula becomes: \[ N(t) = 1000 \times 2^{(t/40)} \] For part b, we want to find the number of bacteria after 60 minutes. Plugging \( t = 60 \) into the formula: \[ N(60) = 1000 \times 2^{(60/40)} = 1000 \times 2^{1.5} \approx 1000 \times 2.83 \approx 2830 \] Thus, after one hour, there will be approximately 2830 bacteria. To answer part c, we set \( N(t) = 50000 \) and solve for \( t \): \[ 50000 = 1000 \times 2^{(t/40)} \] Dividing both sides by 1000: \[ 50 = 2^{(t/40)} \] Taking the logarithm (base 2) of both sides gives us: \[ t/40 = \log_2(50) \] Calculating \( \log_2(50) \) gives approximately 5.64, so: \[ t \approx 5.64 \times 40 \approx 225.5 \text{ minutes} \] Thus, it will take approximately 226 minutes for the culture to grow to 50,000 bacteria. Now for some fun extras! Did you know that the phenomenon of bacterial growth has been observed since the early studies of microbiology in the 19th century? Scientists like Louis Pasteur paved the way for our understanding of how bacteria reproduce and their impact on health and disease. Their work laid the foundation for fields like immunology and microbiome studies, which remain hot topics in research today! For those intrigued by the world of microbes, there are plenty of fascinating books and resources that delve deeper into this subject! “The Hidden Life of Trees” by Peter Wohlleben gives insights into how microorganisms interact with plants, while “Gut: The Inside Story of Our Body's Most Underrated Organ” by Giulia Enders explores the relationship between your gut microbiome and overall health. Happy reading!