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Liliana Romero 12/04/2023 Math 1401 Major: My major is Biomedical Engineering and the research that I would be going off for this equation is the study of the growth of cancer cells in a controlled environment. Studying the growth of cancer cells in a controlled environment is crucial for advancing our understanding of cancer biology and developing effective treatments. By recreating and manipulating the conditions in which cancer cells proliferate in a controlled setting, researchers can investigate the underlying mechanisms of tumor formation, progression, and response to various interventions. This controlled environment allows for precise experiments, enabling scientists to identify key molecular pathways, test potential therapies, and uncover vulnerabilities specific to cancer cells. Problem: A biomedical engineer is researching the growth of cancer cells in a controlled environment. They have developed an equation to model the growth of cancer cells over time: C(t)= 2000 e(0.05t), where C(t) represents the number of cancer cells at time t (in hours) and t is the time in hours. The engineer wants to determine the time it will take for the number of cancer cells to reach a critical threshold for further study. The critical threshold is set at 10,000 cancer cells. The engineer wants to determine the time it will take for the number of cancer cells to reach a critical threshold for further study. The critical threshold is set at 10,000 cancer cells. Question: How many hours will it take for the number of cancer cells to reach the critical threshold of 10,000 in the controlled environment? To solve this problem, the biomedical engineer needs to set up an equation and solve for t when C(t) equals 10,000. Solve: When looking back at our problem we see that we are given the equation C(t)= 2000e(0.05t), we must use this equation to solve for t when C(t)=10,000. When plugging in 10,000 into C(t), our equation would look like 10,000 = 2000e(0.05t). Now we must solve for t. First, you must divide both sides of the equation by 2000 to isolate the exponential term.

10,000/2000 = e(0.05t). Second, you will simplify the left side.10,000 / 2000 = 5. 5= e(0.05t) Third, you need to solve for t. So, you would take the natural logarithm (ln) of both sides of the equation. ln (5) = ln(e(0.05t)). Fourth, by using the property of logarithms it allows you to move the exponent down as a coefficient. ln (5) = 0.05t Finally, solve for t by dividing both sides by 0.05. t = ln (5) / 0.05. Now we must calculate t. t ≈ ln (5) / 0.05 ≈ 13.86 hours (rounded to two decimal places) Conclusion: So, it will take approximately 13.86 hours for the number of cancer cells to reach the critical threshold of 10,000 in the controlled environment. Source: The source of the equation for this problem is the exponential growth model, specifically the formula C(t) = A * e^(rt), where C(t) is the quantity at time t, A is the initial quantity, e is the base of the natural logarithm, r is the growth rate, and t is the time. In this case, the given equation C(t) = 2000e^(0.05t) represents the growth of cancer cells over time, with an initial quantity of 2000 cells, a growth rate of 0.05, and time variable t. The solution process involves algebraic manipulations and the use of logarithmic functions to isolate and solve for the time (t) when the cell count reaches the critical threshold of 10,000. Why the growth model: The specific values in the given problem, such as an initial quantity of 2000 cells and a growth rate of 0.05, are incorporated into the general exponential growth model. This model is commonly used in biology and other fields to describe population growth, decay, or, as in this case, the proliferation of cells over time. The process of solving for C(t) equals 10,000 involves algebraic manipulations and the use of logarithmic functions to isolate and solve for the time at which the critical threshold is reached. Critical Threshold explanation: The critical threshold of 10,000 cancer cells is likely set based on clinical or biological considerations. This threshold may represent a point at which the cancer cell population reaches a critical mass that could lead to observable symptoms, disease progression, or the triggering of certain pathological effects.

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