In an experiment, the population of cells in a culture increase by [tex]$14 \%$[/tex] each hour. If there are 325 cells in the culture now, how many will there be after 24 hours?
[tex]Future Amount = I (1+r)^{ t }[/tex]
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Round to the nearest whole number.
Answer (2)
Identify the initial population I = 325 , the rate of increase r = 0.14 , and the time t = 24 hours.
Substitute these values into the formula: Future Amount = 325 ( 1 + 0.14 ) 24 .
Calculate the future amount: Future Amount = 325 × ( 1.14 ) 24 ≈ 7543.967 .
Round the result to the nearest whole number, which gives the final answer: 7544 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the initial population of cells is 325, and it increases by 14% each hour. We need to find the population after 24 hours using the formula: Future Amount = I ( 1 + r ) t , where I is the initial amount, r is the rate of increase, and t is the time in hours.
Identifying the Variables Now, let's identify the values for each variable:
Initial population, I = 325
Rate of increase, r = 14% = 0.14
Time, t = 24 hours
Substituting the Values Substitute the values into the formula:
Future Amount = 325 ( 1 + 0.14 ) 24
Future Amount = 325 ( 1.14 ) 24
Calculating the Future Amount Calculate the future amount:
Future Amount $= 325 \times (1.14)^{24}
The result of the calculation is approximately 7543.967.
Rounding the Result Round the future amount to the nearest whole number:
Future Amount ≈ 7544
Final Answer Therefore, after 24 hours, there will be approximately 7544 cells in the culture.
Examples
Understanding exponential growth is crucial in various real-world scenarios. For instance, in finance, it helps calculate compound interest on investments. If you invest 1000 a t anann u a l in t eres t r a t eo f 5 = I (1+r)^{ t }$ can determine the investment's value after a certain number of years. Similarly, in epidemiology, this formula can model the spread of infectious diseases, helping predict the number of infected individuals over time, which aids in implementing timely public health measures. Population growth, like the cell culture example, also follows exponential patterns, which is vital for resource planning and environmental management.
After 24 hours, the population of the cells will increase to approximately 7268. This is calculated using the formula for exponential growth, incorporating a 14% hourly increase. The initial population of 325 cells grows significantly due to the continuous increase over 24 hours.
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