Question 32·Medium·Two-Variable Data: Models and Scatterplots
In a physics lab, a metal sphere is heated and then allowed to cool in a room that is maintained at a constant temperature. Measurements show that the sphere’s temperature drops by about 15% of its current temperature each minute during the first several minutes of cooling. What type of model is most appropriate for representing the sphere’s temperature as a function of time during this period?
For model-choice questions, first determine the direction of change (increasing vs. decreasing) to eliminate half the options immediately. Next, focus on how the change is described: a constant amount per unit time points to a linear model (add/subtract the same number each step), while a constant percentage or factor points to an exponential model (multiply by the same number each step). Finally, combine these two ideas—growth vs. decay and linear vs. exponential—to match the situation to the correct model type quickly and confidently.
Hints
Check the direction of change
As time passes, is the sphere’s temperature going up or going down? Use this to immediately rule out some answer choices.
Look carefully at the phrase "15% of its current temperature"
Ask yourself: does this describe a constant amount of change every minute, or a constant percentage of whatever the temperature is at that moment?
Connect constant amount vs. constant percentage to model type
On the SAT, a constant amount of change suggests one type of model, while a constant percentage (multiplying by the same factor each time) suggests another. Which type fits “drops by 15% of its current temperature each minute”?
Desmos Guide
Graph a percentage-based decrease
In an expression line, type something like T(x) = 100*(0.85)^x (you can just type 100*(0.85)^x). This represents a temperature that starts at 100 units and keeps 15% less than the previous value each minute. Look at the graph: it is decreasing and curved, with the steps between points getting smaller over time.
Compare to a linear decrease
In another line, type a linear function such as 100 - 15x. This represents dropping 15 degrees every minute. Compare this straight-line graph to the curve from the first function. Notice that the straight line has equal vertical drops each minute, while the curve’s drops shrink over time because the change is a constant percentage of the current value.
Match the behavior to the answer choice
Think about which graph (the curved one with shrinking drops or the straight line with equal drops) actually matches the phrase "drops by about 15% of its current temperature each minute." On the test, select the answer choice that describes the type of model shown by the curved, decreasing graph with a constant percentage change.
Step-by-step Explanation
Decide whether the temperature is increasing or decreasing
The problem says the sphere is heated and then allowed to cool, and that its temperature drops each minute.
That means as time increases, the temperature must decrease, not increase. So any model that describes an increasing function (like a line with positive slope or a growing curve) cannot match this situation.
Distinguish between linear and exponential change
A linear model means the quantity changes by the same amount every unit of time (for example, “drops by 3 degrees each minute”). This creates a straight-line graph.
An exponential model means the quantity changes by the same percentage (or factor) every unit of time (for example, “drops by 15% of its current value each minute”). This creates a curved graph where the size of the change gets smaller over time if it is decreasing.
The problem states: the temperature drops by about 15% of its current temperature each minute. That is a constant percentage change, not a constant degree change.
So the situation must be modeled by a function where you repeatedly multiply by the same number (like multiplying by each minute), not subtract the same number.
Combine direction (down) with type of change (percentage)
We know two key facts:
- The temperature is going down over time, so the function must represent a decrease, not an increase.
- The change is by a constant percentage of the current value each minute, which is the defining feature of an exponential, not linear, model.
Putting these together, the best description of the model is an exponential decay model.