A company opens an account with an initial balance of $ 36,100.00 . The account earns interest, and no additional

Question

A company opens an account with an initial balance of $ 36,100.00 . The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function A , where A ( t ) is the account balance, in dollars, t years after the account is opened. The account balance after 13 years is $ 68,071.93 . Which equation could define A ?

Accepted Answer

Choice A is correct. Since it's given that the account balance, A t , in dollars, after t years can be modeled by an exponential function, it follows that function A can be written in the form A t = N r t , where N is the initial value of the function and r is a constant related to the growth of the function. It's given that the initial balance of the account is $ 36,100.00 , so it follows that the initial value of the function, or N , must be 36,100.00 . Substituting 36,100.00 for N in the equation A t = N r t yields A t = 36,100.00 r t . It's given that the account balance after 13 years, or when t = 13 , is $ 68,071.93 . It follows that A 13 = 68,071.93 , or 36,100.00 r 13 = 68,071.93 . Dividing each side of the equation 36,100.00 r 13 = 68,071.93 by 36,100.00 yields r 13 = 68,071.93 36,100.00 . Taking the 13 th root of both sides of this equation yields r = 68,071.93 36,100.00 13 , or r is approximately equal to 1.05 . Substituting 1.05 for r in the equation A t = 36,100.00 r t yields A t = 36,100.00 1.05 t , so the equation A t = 36,100.00 1.05 t could define A . Choice B is incorrect. Substituting 0 for t in this function indies an initial balance of $ 31,971.93 , rather than $ 36,100.00 . Choice C is incorrect. Substituting 0 for t in this function indies an initial balance of $ 31,971.93 , rather than $ 36,100.00 . Additionally, this function indies the account balance is decreasing, rather than increasing, over time. Choice D is incorrect. This function indies the account balance is decreasing, rather than increasing, over time.

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