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? ​

Answers

  1. 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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