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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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