Question 54·Hard·Two-Variable Data: Models and Scatterplots
A compound is placed in a solution at time seconds, and its remaining mass is measured every second. The data are shown.
| Time (seconds) | Mass (milligrams) |
|---|---|
| 0 | 200 |
| 1 | 146 |
| 2 | 107 |
| 3 | 78 |
| 4 | 57 |
Which type of model best describes the relationship between time and the remaining mass of the compound?
For SAT questions asking which model type (linear, quadratic, exponential, etc.) best fits a table, quickly test for patterns: check if first differences are roughly constant (linear), if second differences are roughly constant (quadratic), or if the ratios between consecutive values are roughly constant (exponential). You don't need to write full equations—just do a few quick subtractions and divisions, see which pattern looks most consistent, and then pick the model name that matches that pattern and the direction of change (increasing or decreasing).
Hints
Look at how much the mass changes each second
Find the differences between consecutive mass values (for example, , , and so on). Are these differences about the same, or are they changing?
Compare patterns: differences vs. ratios
After you check the differences, try comparing the ratios of consecutive values (for example, , , etc.). Do the ratios show a clearer, more consistent pattern than the differences?
Connect the pattern to a function type
Recall how each model type behaves: linear has a constant change, quadratic has a constant change of the change, and one common model has a roughly constant ratio between outputs for equal time steps. Which option names that last type, with the mass going down over time?
Desmos Guide
Enter the data as a table
In Desmos, add a table and enter the time values in the first column as and the mass values in the second column as . Look at the plotted points to see the overall shape (curved vs. straight).
Numerically compare differences and ratios
In new Desmos lines, type the differences , , , and , and note how they change. Then type the ratios , , , and , and see whether these values stay more nearly constant. Use the observation of which pattern (differences or ratios) is steadier to decide which named model type best matches the data.
Step-by-step Explanation
Check if the decrease is by a constant amount (linear test)
Compute how much the mass changes each second.
- From to :
- From to :
- From to :
- From to :
These differences are not constant, so the relationship is not well modeled by a linear function (which would have the same difference each step).
Check if a quadratic model seems likely (second-difference test)
For a quadratic pattern, the second differences (the differences of the differences) are roughly constant.
First differences (already found): .
Now find how these change:
- Change from to :
- Change from to :
- Change from to :
These second differences are not close to constant, so a quadratic model is unlikely.
Check if the decrease is by a nearly constant percentage (ratio test)
Now compare the ratio of each mass to the previous one.
These ratios are all very close to each other. That means each second, the mass is multiplied by nearly the same factor (about ), which indicates a model where the output changes by a nearly constant percent over equal time intervals.
Match the pattern to the correct model type
On the SAT, you should link patterns to model types:
- Constant difference linear
- Constant second difference quadratic
- Roughly constant ratio (constant percent change) exponential
Here, we found a nearly constant ratio between consecutive masses, and the mass is going down over time. So the best model is a decreasing exponential model, which corresponds to choice C.