Question 230·Medium·Nonlinear Functions
The function
gives the predicted number of bacteria in a laboratory culture hours after a nutrient solution is added, where .
What is the best interpretation of the statement is approximately equal to in this context?
For interpretation questions with function notation, first pin down what the input variable and the function value represent (including their units). Then, when you see something like , read it as "the value of when the time is 36 hours" and attach the correct units to both the input and output. Finally, scan the choices and eliminate any that swap time and quantity, misuse constants from the formula (like growth factors or initial values), or violate the stated domain; the remaining option that keeps input as time and output as the modeled quantity is the correct interpretation.
Hints
Focus on what and mean
Look back at the sentence that defines the function. What does measure? What does represent in terms of the bacteria?
Translate into words
When you see , think: "the value of when ." In this context, what does mean in terms of the situation?
Decide the units of 36 and 316
In the statement is approximately equal to 316, which number is the time, and which number is the number of bacteria? Use the function definition to decide.
Desmos Guide
Enter the function
Type P(t) = 250*(1.04)^(t/6) into Desmos to define the function modeling the bacteria population.
Evaluate the function at 36 hours
On a new line, type P(36) and look at the output value Desmos gives. Interpret that number as the predicted number of bacteria when the time is 36 hours after the solution was added.
Step-by-step Explanation
Identify what the variables represent
From the description, is the number of hours after the nutrient solution is added, and is the predicted number of bacteria in the culture at that time.
So:
- Input (): time, measured in hours
- Output (): number of bacteria (a count, not a time)
Interpret the meaning of
The expression means we are evaluating the function when .
In words, is: the predicted number of bacteria in the culture 36 hours after the nutrient solution is added.
Notice that 36 must be a time in hours, not a number of bacteria.
Attach the approximate value and units
We are told that is approximately equal to .
- The value is the output of , so its units must be bacteria.
- So means: after 36 hours, the model predicts about 316 bacteria in the culture.
Now match this to the answer choices: this corresponds to “The predicted number of bacteria in the culture is about 316 after 36 hours since the solution was added.”