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Advanced Math / Nonlinear functions Difficulty: Medium

Time (years)	Total amount (dollars)
$0$ $0$	$604.00$ $604.00$
$1$ $1$	$606.42$ $606.42$
$2$ $2$	$608.84$ $608.84$

Rosa opened a savings account at a bank. The table shows the exponential relationship between the time $t$ , in years, since Rosa opened the account and the total amount $n$ , in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between $t$ and $n$ ?

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Explanation

Choice C is correct. It’s given that the relationship between $t$ and $n$ is exponential. The table shows that the value of $n$ increases as the value of $t$ increases. Therefore, the relationship between $t$ and $n$ can be represented by an increasing exponential equation of the form $n = a {(1 + b)}^{t}$ , where $a$ and $b$ are positive constants. The table shows that when $t equals 0$ , $n equals 604$ . Substituting $0$ for $t$ and $604$ for $n$ in the equation $n = a {(1 + b)}^{t}$ yields $604 = a {(1 + b)}^{0}$ , which is equivalent to $604 equals a left parenthesis 1 right parenthesis$ , or $604 equals a$ . Substituting $604$ for $a$ in the equation $n = a {(1 + b)}^{t}$ yields $n = 604 {(1 + b)}^{t}$ . The table also shows that when $t equals 1$ , $n equals 606.42$ . Substituting $1$ for $t$ and $606.42$ for $n$ in the equation $n = 604 {(1 + b)}^{t}$ yields $606.42 = 604 {(1 + b)}^{1}$ , or $606.42 = 604 (1 + b)$ . Dividing both sides of this equation by $604$ yields approximately $1.004 = 1 + b$ . Subtracting $1$ from both sides of this equation yields that the value of $b$ is approximately $0.004$ . Substituting $0.004$ for $b$ in the equation $n = 604 {(1 + b)}^{t}$ yields $n = 604 {(1 + 0.004)}^{t}$ . Therefore, of the choices, choice C best represents the relationship between $t$ and $n$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.