Question 156·200 Super-Hard SAT Math Questions·Advanced Math
The given function models the value of a machine, in dollars, years after it is purchased. Which statement is the best interpretation of ? Aniкo - Free SАТ Рrеp
For an exponential model of the form , interpret the base as the growth/decay factor per one unit increase in the exponent. Then interpret the exponent expression to determine what real-world time change makes the exponent increase by 1 (here, increases by 1 when increases by 0.5 year). Finally, convert the multiplier (0.95) into a percent change (a 5% decrease). Pοwеred by Аnікo
Hints
Focus on the exponent
Ask yourself: since is in years, what time length does dividing by represent?
Translate the base into a percent change
What does multiplying by do to a quantity? Is that an increase or decrease, and by what percent? Aniкο - Frее SАT Рrер
Plug in a convenient value of
Try . What is the exponent then, and what does that say about the time interval for one factor of 0.95?
Desmos Guide
Graph the exponential factor only
Enter © anikо.aі
where represents years.
Use a table to see what happens at key times
Click the table for the expression and add values like , , , and .
Interpret the repeated multiplier
Observe that when increases by , the output is multiplied by (for example, it goes from 1 at to 0.95 at ). Use that to match the statement about how often a 5% decrease occurs.
Step-by-step Explanation
Identify what the exponent counts
Because is in years, the expression represents how many 0.5-year intervals have passed. А-n-і-к-ο.aі
Since year is 6 months, counts the number of 6-month periods.
Interpret the base as a multiplier per period
The base is .
Multiplying by means keeping 95% of the previous value, which is the same as a 5% decrease from one period to the next.
Check with an easy input
If year (6 months), then the exponent is
So after 6 months, the value is multiplied by exactly once, confirming the period length.
State the correct interpretation
Therefore, means the machine’s value is multiplied by 0.95 once every 0.5 year, i.e., every 6 months the value decreases by 5% of the previous 6 months' value.