Question 116·Medium·Nonlinear Functions
A scientist is monitoring the temperature, in degrees Fahrenheit, of a cup of coffee as it cools. The temperature , in degrees Fahrenheit, after minutes is modeled by
Which statement is the best interpretation of the constant in this context?
For exponential growth and decay models on the SAT, first identify the structure: a constant term plus an exponential term, . Quickly compute to find the initial amount, and check whether any constant in the formula matches that value. Then think about what happens as becomes very large: the exponential part tends to 0, so the function approaches the constant . Interpreting that long-term value in context (like a stable or ambient level) usually reveals what the constant represents, and prevents confusing it with a rate of change or a value at a specific time.
Hints
Look at the initial value
Substitute into . What number do you get, and which option (if any) matches that number?
Check the value after 1 minute
Now substitute into the formula. Is that value equal to , or something else?
Think about long-term behavior
As gets very large, what happens to ? What temperature will the coffee get closer and closer to over a long time?
Connect the math to the real situation
In real life, when a hot drink cools down for a long time, what temperature does it approach and stop changing much beyond?
Desmos Guide
Enter the function
In Desmos, type T(x) = 70 + 130*(0.85)^x so that represents the time in minutes and is the temperature.
Check specific values
Use the table feature or just type T(0) and T(1) into Desmos. Compare these values with 70 to see whether 70 matches the initial temperature or the temperature after 1 minute.
Observe the graph for large times
Look at the graph of for larger values (for example, from to ). Notice what -value the graph gets closer and closer to without going below.
Add a reference line
Type y = 70 as a separate graph. See how the curve of approaches this horizontal line as increases, and think about what constant temperature that line represents in the real situation.
Step-by-step Explanation
Understand the structure of the function
The temperature is modeled by
This has two parts:
- A constant term, .
- A term that changes with time, .
In exponential models of this type, the constant outside the exponential usually represents a baseline or limiting value that the function approaches as time increases.
Check what happens at time t = 0
To see whether could be the initial temperature, calculate :
So the initial temperature of the coffee is degrees Fahrenheit, not . That eliminates any interpretation that says is the initial temperature.
Check the other simple time and the rate idea
Now check :
So is about degrees, not , which means is not the temperature after 1 minute either.
Also, in an exponential expression like :
- The is the factor the temperature difference is multiplied by each minute.
- The is the initial difference between the coffee and the baseline .
Neither of these makes an amount of change per minute.
Interpret the constant 70 in the real situation
As time goes on, the factor gets smaller and smaller and approaches . That means the whole term approaches , and the temperature approaches
So is the temperature the coffee cools toward and eventually levels off at. In real life, a cooling object approaches the temperature of its surroundings, so the constant represents the temperature of the surrounding room.