Question 31·Easy·Two-Variable Data: Models and Scatterplots
A radioactive isotope has an initial mass of 800 grams and decays by 12% each day. What type of function best models the relationship between the isotope's remaining mass and the number of days since the start of observation?
When a SAT question asks what type of function models a situation, first decide whether the quantity increases or decreases over time by looking for words like "grows" or "decays." Next, check whether each equal time step changes the quantity by a fixed amount (linear) or a fixed percentage/factor (exponential); you can test this by imagining the first couple of steps and seeing if you are adding/subtracting the same number or multiplying by the same number. Finally, match this behavior to the choices: straight-line increase/decrease means linear, while a curved increase/decrease from repeated multiplication means exponential, and then pick whether it is increasing or decreasing.
Hints
Check the direction of change
The problem says the isotope "decays" by 12% each day. Does that mean the mass is going up or going down as days pass? Which answer choices describe functions that go down over time?
Constant amount or constant percentage?
Ask yourself: Is the mass losing the same number of grams every day, or the same percentage of whatever mass is left? That difference tells you whether the graph would be a straight line or a curve.
Imagine the first few days
Start with 800 grams. After 1 day, find 88% of 800 (since 12% decays). After 2 days, do you subtract the same number of grams again, or do you again take 88% of the new amount? Use that pattern to decide which function type matches the situation.
Desmos Guide
Graph the percent-decay model
In Desmos, type the equation y = 800*(0.88)^x. This represents starting at 800 grams and multiplying by 0.88 each day (for x days). Look at how the graph behaves as increases: it should start at 800 and move downward while curving and getting closer to 0.
Compare with a straight-line decrease
For comparison, also type y = 800 - 96x, which would represent losing 96 grams (12% of 800) every day. This graph is a straight line that goes down at a constant rate.
Match the graph to the answer choices
Compare the two graphs: the realistic model of radioactive decay is the one that curves downward and levels off above 0, not the straight line or any graph that goes upward. Choose the answer choice that describes a function that decreases over time and follows a curved pattern like the first graph.
Step-by-step Explanation
Interpret "decays by 12% each day"
The phrase "decays by 12% each day" means that each day the isotope keeps 88% of whatever mass it has at the start of that day.
- 12% as a decimal is .
- The remaining fraction each day is .
- After 1 day, the mass is grams.
- After 2 days, you again take 88% of the new mass: .
So every day you are multiplying the current mass by .
Decide if the pattern is linear or exponential
A linear model changes by the same fixed amount each step (for example, always losing 96 grams per day).
Here, the change is a fixed percentage of the current amount, not a fixed number of grams:
- Day 1 loss: grams
- Day 2 loss: grams (smaller than 96)
Because you are repeatedly multiplying by the same factor , the relationship fits an exponential pattern of the form , where is the number of days.
Determine whether the exponential model increases or decreases
From the context ("radioactive" and "decays") and the calculations, the mass gets smaller each day.
- The factor you multiply by each day is , which is less than 1, so the values go down over time.
- The model would look like
which is an exponential function that decreases as increases.
So the situation is best described by a decreasing exponential function, making Decreasing exponential the correct answer.