The registration period for the race is under way, and as the word is getting out, the number of participants registering is gro
wing exponentially. The numbers from the first two days of registration are shown in the table.
Use the data in the table and the general form of an exponential growth equation, y= a(1 +r)^x, to write an equation modeling the exponential growth of registrations, where x is the number of days since registration opened, a is the initial number of registrations, r is the rate of growth of registrations per day, and y is the total number of registrations.
Use the data in the table and the general form of an exponential growth equation, y= a(1 +r)^x, to write an equation modeling the exponential growth of registrations, where x is the number of days since registration opened, a is the initial number of registrations, r is the rate of growth of registrations per day, and y is the total number of registrations.
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If we use the substitution 
, then 
. If you try substituting just 
and 
into the integrand, though, you'll notice that there's a 
left over that we have to deal with.
To get rid of this problem, use the identity
and substitute in the left side of the identity for the extra , as shown:
From there, we can substitute in and , and then evaluate:
To get rid of this problem, use the identity
From there, we can substitute in