Question 14·Medium·Two-Variable Data: Models and Scatterplots
A chemist measures the mass of a radioactive isotope that begins with 200 milligrams at noon and loses about 7% of its remaining mass every hour. What type of model is most appropriate to represent the mass of the isotope as a function of time since noon?
When a question asks for the “type of model,” first decide if the change is by a constant amount (linear) or a constant percentage/factor (exponential). Look for words like “each hour,” “each year,” and especially a percent of the current amount to signal exponential behavior. Then check whether the quantity is going up or down over time: increasing means positive slope or growth; decreasing means negative slope or decay. Matching both the pattern (amount vs. percent) and the direction (increase vs. decrease) lets you quickly eliminate wrong choices and pick the correct model.
Hints
Focus on the type of change each hour
Is the isotope losing the same number of milligrams every hour, or the same percentage of its remaining mass every hour?
Think about linear vs. exponential patterns
Linear models change by adding or subtracting a fixed amount each time. Exponential models change by multiplying by a fixed factor (often involving a percent). Which one fits “loses 7% of its remaining mass every hour”?
Decide if the function should increase or decrease
Over time, is the mass getting larger or smaller? Once you know if it’s increasing or decreasing, look for the option that matches both the pattern of change (amount vs. percent) and the direction (up vs. down).
Desmos Guide
Enter an expression that matches the description
In Desmos, type 200*(0.93)^x. This expression starts at 200 when and multiplies the current value by for each increase of 1 in , just like losing 7% of the remaining mass every hour.
Observe the shape of the graph
Look at the curve of 200*(0.93)^x. Notice that it decreases as increases and that the graph is curved, not a straight line. This shows a repeated-percent decrease over time.
Compare with a typical linear model
Now type a linear example like 200-14x. This line goes down at a constant rate and appears as a straight line. Compare it to 200*(0.93)^x to see the difference between constant-amount decrease (linear) and curved, percentage-based decrease (the model described in the problem). Use this comparison to decide which answer choice best matches the situation.
Step-by-step Explanation
Restate the situation in your own words
The isotope starts with 200 milligrams at noon. Every hour, it loses 7% of whatever mass it has at that moment. So after each hour, it keeps 93% of its current mass.
Decide between linear and exponential behavior
Ask: Is the change each hour a constant amount or a constant percentage?
- Linear models have a constant amount of change each step (like “minus 5 mg every hour”).
- Exponential models have a constant percentage change each step (like “loses 7% every hour”).
Here, the problem says it loses 7% of its remaining mass every hour, which is a constant percentage, not a constant amount. That tells you the situation is best modeled by an exponential function, not a linear one.
Recognize the exponential factor
If is the mass (in milligrams) after hours, then keeping 93% each hour means you multiply by each hour.
A model that matches the description would look like:
This has the typical exponential form (starting amount and repeated multiplication by ), which confirms it is exponential rather than linear.
Determine whether it represents growth or decay
In an exponential model :
- If , the quantity grows over time (exponential growth).
- If , the quantity shrinks over time (exponential decay).
Here, , which is less than 1, and the mass is clearly decreasing every hour. So the appropriate choice is an exponential decay model.