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Advanced Math / Nonlinear functions Difficulty: Medium

A company opens an account with an initial balance of $dollar sign 36,100.00$ . 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 $dollar sign 68,071.93$ . Which equation could define $A$ ?

$upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t$

$upper A left parenthesis t right parenthesis equals 31,971.93 left parenthesis 1.05 right parenthesis Superscript t$

$upper A left parenthesis t right parenthesis equals 31,971.93 left parenthesis 0.05 right parenthesis Superscript t$

$upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 0.05 right parenthesis Superscript t$

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Explanation

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 $dollar sign 36,100.00$ , 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 $upper A left parenthesis t right parenthesis equals 36,100.00 r Superscript t$ . It's given that the account balance after $13$ years, or when $t equals 13$ , is $dollar sign 68,071.93$ . It follows that $upper A left parenthesis 13 right parenthesis equals 68,071.93$ , or $36,100.00 r Superscript 13 Baseline equals 68,071.93$ . Dividing each side of the equation $36,100.00 r Superscript 13 Baseline equals 68,071.93$ by $36,100.00$ yields $r^{13} = \frac{68,071.93}{36,100.00}$ . Taking the $13$ th root of both sides of this equation yields $r = \sqrt[13]{\frac{68,071.93}{36,100.00}}$ , or $r$ is approximately equal to $1.05$ . Substituting $1.05$ for $r$ in the equation $upper A left parenthesis t right parenthesis equals 36,100.00 r Superscript t$ yields $upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t$ , so the equation $upper A left parenthesis t right parenthesis equals 36,100.00 left parenthesis 1.05 right parenthesis Superscript t$ could define $A$ .

Choice B is incorrect. Substituting $0$ for $t$ in this function indicates an initial balance of $dollar sign 31,971.93$ , rather than $dollar sign 36,100.00$ .

Choice C is incorrect. Substituting $0$ for $t$ in this function indicates an initial balance of $dollar sign 31,971.93$ , rather than $dollar sign 36,100.00$ . Additionally, this function indicates the account balance is decreasing, rather than increasing, over time.

Choice D is incorrect. This function indicates the account balance is decreasing, rather than increasing, over time.