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?
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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) = aebtln(y(t)) = ln(aebt)ln(y(t)) = ln(a) + ln(ebt)ln(y(t)) = ln(a) + btNow, 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) + 2bAfter 4 days, there are 400 files, so we can use the following equation to solve for b:ln(400) = ln(a) + 4bWe can solve this system of equations using elimination method. Subtracting the first equation from the second equation gives:ln(400) – ln(200) = 4b – 2bln(2) = 2bb = ln(2)/2Now 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) + 2bln(200) = ln(a) + 2(ln(2)/2)ln(200) = ln(a) + ln(2)ln(200) – ln(2) = ln(a)ln(100) = ln(a)a = 100So 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) = 100e2.079y(6) = 691.6Therefore, after 6 days, there will be approximately 691.6 files.