Question 24·Hard·Two-Variable Data: Models and Scatterplots
A chemist models the mass , in milligrams, of a radioactive sample minutes after it is prepared with the equation
Which choice best interprets the value in this context?
For exponential models on the SAT, read the coefficient as the starting value and the base as the multiplicative change for each 1-unit increase in the independent variable. If the base is less than 1, the quantity is decaying, so convert the base to a percent to decide how much remains each step rather than treating it like a constant amount subtracted each time.
Hints
Look at the structure of the model
In an exponential model of the form , focus on what the coefficient means and what the base means.
Track what happens when time increases by 1
Ask what happens to the expression when changes from 0 to 1. Which part of the formula tells you how the mass changes from one minute to the next?
Convert the factor to a percent
The number can be read as a percent of the previous amount. Decide whether it describes the amount that remains or the amount that is lost.
Desmos Guide
Enter the model
In Desmos, enter y=4.6(0.971)^x and open a table for the expression so you can compare the values at nearby times.
Compare consecutive values
Look at the outputs when and . The value at is times the value at , so the base tells you the factor applied each minute.
Translate the factor into words
Because the graph multiplies by for each 1-minute increase, the correct interpretation is that each minute, the sample has 97.1% of the mass it had 1 minute earlier.
Step-by-step Explanation
Identify the role of the base
The model is
In an exponential expression of the form , the coefficient is the initial amount and the base is the factor applied each time increases by 1.
Interpret the factor
Here, the base is . That means whenever goes up by 1 minute, the mass is multiplied by .
Since , the sample keeps 97.1% of its previous mass each minute.
Match the interpretation to a choice
The correct statement is that each minute, the sample has 97.1% of the mass it had 1 minute earlier.
So the correct answer is Each minute, the sample has 97.1% of the mass it had 1 minute earlier.