Question 1104549: Please help me solve this word problem: in 1997, there were a total of 1,316,999 bankruptcies filed under the Bankruptcy Reform Act. The model for the number of bankruptcies filed is B(t)=0.798*1.164^t, where t is the number of years since 1994 and B is the number of bankruptcies filed in terms of millions (administrative office in US courts, statistical tables for the federal judiciary). In what year is it predicted that 12 million bankruptcies will be filed?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
you are told that there were 1,316,999 bankruptcies some time in 1997.
the exact time in 1997 is not specified, so it could be anywhere between january 1, 1997 and december 31, 1997.
you are told that T is equal to the number of years since 1994.
assuming that the year indicated means january 1st of the year indicated, then T = 3 means january 1, 1997 and T = 4 means january 1, 1998.
you can confirm that there will be 1,316,999 bankruptcies sometime in 1997 by applying the formula for T = 3 and T = 4.
when T = 3, the number of bankruptcies is equal to 1.258524957 million.
whena T = 4, the number of bankruptcies is equal to 1.46492305 million.
1.316999 million is somewhere between them, therfore, the number of bankruptcies will reach 1.316999 million sometime between january 1, 1997 and january 1, 1998, which is sometime in the year 1997.
to predict the year when the number of bankruptcies will reach 12 million, the formula becomes 12 = .798 * 1.164 ^ T.
divide both sides of this formula by .798 to get 12/.798 = 1.164 ^ T.
take the log of both sides of this equation to get log(12/.798) = log(1.164 ^ T).
since log(1.164 ^ T is equal to T * log(1.164), your formula becomes log(12/.798) = T * log(1.164).
divide both sides of this formula by log(1.164) to get log(12/.798) / log(1.164) = T
solve for T to get T = 17.84875148.
add 1994 to that to get the year = 2011.848751.
since T = 17 is equal to january 1, 2011 and T = 18 is equal to january 1,2012, then the number of bankruptcies will reach 12,000,000 some time in the year 2011.
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