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Advanced Math / Nonlinear functions Difficulty: Medium

Time (years)	Total amount (dollars)
$0$ $0$	$670.00$ $670.00$
$1$ $1$	$674.02$ $674.02$
$2$ $2$	$678.06$ $678.06$

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

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Explanation

Choice B is correct. It’s given that the relationship between $t$ and $d$ is exponential. The table shows that the value of $d$ increases as the value of $t$ increases. Therefore, the relationship between $t$ and $d$ can be represented by an increasing exponential equation of the form $d = a {(1 + b)}^{t}$ , where $a$ and $b$ are positive constants. The table shows that when $t equals 0$ , $d equals 670$ . Substituting $0$ for $t$ and $670$ for $d$ in the equation $d = a {(1 + b)}^{t}$ yields $670 = a {(1 + b)}^{0}$ , which is equivalent to $670 equals a left parenthesis 1 right parenthesis$ , or $670 equals a$ . Substituting $670$ for $a$ in the equation $d = a {(1 + b)}^{t}$ yields $d = 670 {(1 + b)}^{t}$ . The table also shows that when $t equals 1$ , $d equals 674.02$ . Substituting $1$ for $t$ and $674.02$ for $d$ in the equation $d = 670 {(1 + b)}^{t}$ yields $674.02 = 670 {(1 + b)}^{1}$ , or $674.02 = 670 (1 + b)$ . Dividing both sides of this equation by $670$ yields $1.006 = 1 + b$ . Subtracting $1$ from both sides of this equation yields $b equals 0.006$ . Substituting $0.006$ for $b$ in the equation $d = 670 {(1 + b)}^{t}$ yields $d = 670 {(1 + 0.006)}^{t}$ . Therefore, of the choices, choice B best represents the relationship between $t$ and $d$ .

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

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

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