A company models the value , in dollars, of a piece of equipment years after purchase with the function Which of the following best interprets the factor in this context?

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SAT Nonlinear Functions Question 176 | Free SAT Prep | Aniko

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Question 176·Medium·Nonlinear Functions

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A company models the value $V$ , in dollars, of a piece of equipment $t$ years after purchase with the function

V(t)=2{,}400(0.70)^t.

Which of the following best interprets the factor $0.70$ in this context?

For exponential word problems on the SAT, first identify the initial value (the coefficient) and the growth/decay factor (the base of the exponent). Check whether the base is greater than 1 (growth) or between 0 and 1 (decay), then rewrite the base as a percentage to see how much of the quantity is kept each period and how much is lost or gained. If you’re unsure, plug in $t=0$ and $t=1$ to see how the value changes in one time step and match that change to the answer choices.

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Look at the structure of the function

Focus on $V(t)=2400(0.70)^t$ . In an expression of the form $(\text{initial value})\times(\text{number})^t$ , what does the base (here $0.70$ ) usually represent in exponential models?

Try specific values of t

Find $V(0)$ and $V(1)$ . How does the value after 1 year compare to the value at purchase? What operation connects $V(1)$ to $V(0)$ ?

Convert the decimal to a percentage

Rewrite $0.70$ as a percentage. Then decide whether that percentage represents the portion the equipment keeps or the portion it loses each year.

Desmos Guide

Enter the value function

In Desmos, type V(t) = 2400*(0.70)^t to graph the model or see the expression.

Use a table to see how the value changes

Click the gear icon next to the expression and choose "Convert to table" (or manually add a table with t in the first column and V in the second), then look at the values of $V$ when $t=0$ and $t=1$ .

Compare year 0 and year 1

Compare $V(1)$ to $V(0)$ in the table. Notice what number you would multiply $V(0)$ by to get $V(1)$ , convert that number to a percentage, and then choose the option that describes this yearly change in words.

Step-by-step Explanation

Identify what each part of the function represents

The function is $V(t)=2400(0.70)^t$ .

The number $2400$ is the initial value of the equipment when $t=0$ (the purchase price).
The factor $0.70$ is the amount the value is multiplied by each year. Your goal is to interpret what “multiply by $0.70$ each year” means in words about the equipment’s value.

Compare the value at purchase and after 1 year

Compute the value at $t=0$ and $t=1$ :

At $t=0$ : $V(0)=2400(0.70)^0=2400(1)=2400$ .
At $t=1$ : $V(1)=2400(0.70)^1=2400(0.70)=1680$ . Now compare: $\dfrac{V(1)}{V(0)}=\dfrac{1680}{2400}=0.70$ , which is $70\%$ . So after one year, the new value is $70\%$ of the previous value.

Translate the multiplier into a verbal description

Since each year the value is multiplied by $0.70$ (that is, becomes $70\%$ of what it was), the equipment’s value goes down, but not by $70\%$ .

It keeps $70\%$ of its value each year and therefore loses $30\%$ of its value each year. Among the choices, the statement that matches this is: The equipment retains 70% of its value each year.

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Step-by-step Explanation

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Step-by-step Explanation