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 ?
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).
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?
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
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.
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.