In a research experiment a population of files is increasing according to the law of exponential growth y(t) = aebt where a and b

Question

In a research experiment a population of files is increasing according to the law of exponential growth y(t) = aebt where a and b are constant and t is the number of days after 2 days there are 200 files and after 4 days there are 400 files. How many flies will there be after 6 days?

thienthanh · Answer

Answer:   After 6 days, there will be approximately  691.6 files .   Step-by-step explanation:   We can use the given information to solve for the constants a and b in the exponential growth equation. First, we can rearrange the equation to solve for b:  y(t) = aebt  ln(y(t)) = ln(aebt)  ln(y(t)) = ln(a) + ln(ebt)  ln(y(t)) = ln(a) + bt    Now, we can use the information given in the problem to solve for b. After 2 days, there are 200 files, so we can use the following equation to solve for b:    ln(200) = ln(a) + 2b    After 4 days, there are 400 files, so we can use the following equation to solve for b:    ln(400) = ln(a) + 4b    We can solve this system of equations using elimination method. Subtracting the first equation from the second equation gives:    ln(400) - ln(200) = 4b - 2b  ln(2) = 2b  b = ln(2)/2    Now that we have found the value of b, we can plug it back into one of the original equations to solve for a. Let's use the equation for 2 days:    ln(200) = ln(a) + 2b  ln(200) = ln(a) + 2(ln(2)/2)  ln(200) = ln(a) + ln(2)  ln(200) - ln(2) = ln(a)  ln(100) = ln(a)  a = 100    So now we have the values of a and b, and we can use them to find the number of files after 6 days:    y(6) = 100e(ln(2)/2)(6)  y(6) = 100e(3ln(2))  y(6) = 100e(3 * 0.693)  y(6) = 100e2.079  y(6) = 691.6    Therefore, after 6 days, there will be approximately 691.6 files.

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