Question 702·Medium·Math

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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?

In an exponential model, the factor multiplies the value each time the exponent increases by $1$ . If the exponent counts years, the factor applies once per year. For a factor below $1$ , distinguish the yearly multiplier from the percent decrease. Subtract from $100\%$ only when you need the percent decrease; you don't need to calculate any dollar values.

Hints

Hint 1
In a model of the form $a(b)^t$ , the factor $b$ is used each time $t$ increases by $1$ . Since $t$ counts years, how often does the factor apply?
Hint 2
A factor below $1$ is a fraction of the previous value, not a fixed dollar amount. Convert the decimal to a percent, then compare it with the whole previous value, $100\%$ , to distinguish what remains from what is lost.

Step-by-step

Read the yearly factor
Step 1
Identify what happens each year
No Desmos needed. The question asks you to interpret the factor, not calculate a value. In this exponential model, $t$ counts years. Each time $t$ increases by $1$ , the value gets another factor of $0.70$ , so $V(t+1)=0.70V(t)$ . The same multiplier applies each year.
Step 2
Convert the factor to a percent
Convert the yearly factor to a percent so you can compare it with the previous year's full value: $0.70=70\%$ .
Step 3
Separate what remains from what is lost
The part lost is the rest of the previous value: $100\%-70\%=30\%$ . The factor describes what remains, not what was taken away. So the equipment retains $70\%$ of its value each year, rather than losing $70\%$ . Choice D.

Question 702·Medium·Math

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