Problem 89
Question
BIOMEDICAL: Cell Growth One leukemic cell in an otherwise healthy mouse will divide into two cells every 12 hours, so that after \(x\) days the number of leukemic cells will be \(f(x)=4^{x}\). a. Find the approximate number of leukemic cells after 10 days. b. If the mouse will die when its body has a billion leukemic cells, will it survive beyond day \(15 ?\)
Step-by-Step Solution
Verified Answer
a. Approximately 1,048,576 cells after 10 days. b. The mouse will not survive beyond day 15.
1Step 1: Understanding the Cell Growth Function
The number of leukemic cells after \(x\) days is given by the function \(f(x) = 4^x\). This exponential function doubles the number of cells every 12 hours, or quadruples it every day.
2Step 2: Calculating the Number of Cells After 10 Days
To find the number of leukemic cells after 10 days, we evaluate the function \(f(x)\) at \(x = 10\): \[f(10) = 4^{10}\]. Using a calculator, we find that \(4^{10} = 1048576\). There will be approximately 1,048,576 leukemic cells after 10 days.
3Step 3: Calculating the Number of Cells After 15 Days
To determine the number of cells after 15 days, evaluate \(f(x)\) at \(x = 15\): \[f(15) = 4^{15}\]. Calculating this gives \(4^{15} = 1073741824\) cells.
4Step 4: Comparing to the Survival Threshold
A mouse will die when the number of cells reaches a billion (\(1,000,000,000\)). Since \(1073741824 > 1,000,000,000\), the number of cells after 15 days exceeds the lethal threshold.
Key Concepts
Leukemic CellsCell Growth FunctionSurvival Threshold
Leukemic Cells
Leukemic cells are cancerous white blood cells that grow uncontrollably, impairing the body's ability to fight infections and perform normal physiological functions. In the context of exponential growth, a single leukemic cell in an otherwise healthy organism can rapidly multiply. This rapid multiplication is a characteristic of cancer cells due to their ability to bypass usual cell cycle checkpoints, leading to unchecked proliferation.
For example, in the case of leukemic cells in a mouse, we observe that one leukemic cell doubles every 12 hours. This means, within a single day (24 hours), each cell cycle results in four times the number of cells compared to the day before. This rapid proliferation is a typical outcome of exponential growth, where the quantity increases by a consistent multiple over equal successive periods.
For example, in the case of leukemic cells in a mouse, we observe that one leukemic cell doubles every 12 hours. This means, within a single day (24 hours), each cell cycle results in four times the number of cells compared to the day before. This rapid proliferation is a typical outcome of exponential growth, where the quantity increases by a consistent multiple over equal successive periods.
Cell Growth Function
A cell growth function describes how the number of cells changes over time. For leukemic cells in our problem, the growth is modeled by the function \( f(x) = 4^x \), where \( x \) represents the number of days.
This function is an example of exponential growth, where the number of cells increases by a power of four for each day. Understanding how exponential functions work is key in biology and medical studies, as many populations of cells—both healthy and unhealthy—can follow similar growth patterns.
Important points about exponential growth in cell populations include:
This function is an example of exponential growth, where the number of cells increases by a power of four for each day. Understanding how exponential functions work is key in biology and medical studies, as many populations of cells—both healthy and unhealthy—can follow similar growth patterns.
Important points about exponential growth in cell populations include:
- Growth Rate: The number of cells increases at a rate proportional to the current number, leading to rapid increases over time.
- Doubling Time: The period needed for the number of cells to double is constant in exponential growth. In this case, cells double every 12 hours.
Survival Threshold
The survival threshold in our exercise refers to the maximum number of leukemic cells a mouse can sustain before it reaches a critical point where survival is not possible. In this example, this threshold is set at one billion leukemic cells.
This concept is crucial in understanding the severity of diseases like leukemia. As the number of leukemic cells crosses this threshold, the body's ability to function normally deteriorates, leading to the eventual death of the organism.
Important considerations about survival thresholds include:
This concept is crucial in understanding the severity of diseases like leukemia. As the number of leukemic cells crosses this threshold, the body's ability to function normally deteriorates, leading to the eventual death of the organism.
Important considerations about survival thresholds include:
- Critical Limits: Different organisms have varying tolerances to a high number of abnormal cells based on size, health, and immune response.
- Disease Progression: As leukemic cells grow, they interfere with normal biological functions much earlier than the lethal threshold, showcasing the body's declining health over time.
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