The value , in dollars, of a vintage guitar collection is modeled by the equation where is the number of years since 2005. What is the best interpretation of the number in this context?

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SAT Nonlinear Functions Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Nonlinear Functions

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The value $V$ , in dollars, of a vintage guitar collection is modeled by the equation

V = 12,500(1.06)^t

where $t$ is the number of years since 2005.

What is the best interpretation of the number $1.06$ in this context?

For questions about interpreting parameters in exponential models, first rewrite the expression in the standard form $V = a(b)^t$ and label $a$ as the initial value (when $t=0$ ) and $b$ as the constant growth/decay factor applied each time $t$ increases by 1. Check units (dollars, percent, or unitless multiplier) and whether $b$ is greater than or less than 1 to decide if it represents growth or decay, and then choose the option whose wording matches that specific role, avoiding answers that treat a multiplier as a dollar amount or that mix up increase with decrease.

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Identify the roles of the numbers in the formula

Focus on the structure $V = 12{,}500(1.06)^t$ . In an expression of the form $a(b)^t$ , what does $a$ usually represent? What does $b$ usually represent?

Check what happens at $t=0$ and $t=1$

Plug in $t=0$ and $t=1$ to see what $V$ equals in each case. Which number gives the value in 2005, and which number controls how $V$ changes from year to year?

Think about units and increase vs. decrease

Does $1.06$ look like a dollar amount, a percent, or a unitless multiplier? Is it greater than 1 or less than 1, and does that match an increase or a decrease each year?

Connect the base of the exponent to yearly change

In an exponential model, how does the value change when $t$ increases by 1? What role does the base of the exponent (here, $1.06$ ) play in that change?

Desmos Guide

Enter the exponential model

Type V = 12500*(1.06)^t into Desmos. Desmos may treat this as a function; if so, write V(t) = 12500*(1.06)^t instead.

Compare values for consecutive years

Create a table for the function (click the plus sign and choose "table"), and let $t$ take values like 0, 1, 2. Observe the values $V(0)$ , $V(1)$ , $V(2)$ , and look at the ratio $V(1)/V(0)$ and $V(2)/V(1)$ to see how the value changes each time $t$ increases by 1.

Connect the ratio to the base

Notice that the ratio of the value from one year to the previous year is constant and matches the base in the equation. Use this observation to decide which answer choice correctly describes the role of that base number in the context of the problem.

Step-by-step Explanation

Recognize the exponential growth form

The equation is

V = 12{,}500(1.06)^t

This matches the general exponential model $V = a(b)^t$ , where:

$a$ is the initial value (when $t=0$ ), and
$b$ is the constant number you multiply by each time $t$ increases by 1 (each year, in this problem).

Identify what 12,500 and 1.06 represent

Set $t=0$ :

V = 12{,}500(1.06)^0 = 12{,}500(1) = 12{,}500

So $12{,}500$ is the value of the collection in 2005. That means $1.06$ is not the initial value and not in dollars; instead, it is the base of the exponent, the number the value is repeatedly multiplied by each year.

Relate 1.06 to yearly change

Compare the value after one year to the initial value:

At $t=0$ , $V_0 = 12{,}500$ .
At $t=1$ , $V_1 = 12{,}500(1.06)$ .

That means

\frac{V_1}{V_0} = \frac{12{,}500(1.06)}{12{,}500} = 1.06

So each year, the collection’s value becomes $1.06$ times what it was the previous year—this is a $6\%$ increase, expressed as a multiplier.

Match the math meaning to the answer choice

Because $1.06$ is the constant number you multiply the collection’s value by each year (a growth factor of $1.06$ per year), the correct interpretation is:

D) The factor by which the value of the collection multiplies each year.

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Step-by-step Explanation

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Step-by-step Explanation