Question 161·Medium·Nonlinear Functions
A dose of a radioactive tracer is administered at noon. The amount of the tracer, in milligrams, remaining in the body hours after noon is modeled by the function
Which of the following is the best interpretation of “ is approximately equal to ” in this context?
For interpretation questions with functions, first identify what the input and output represent (including units). Then translate a value like into plain language using the pattern "[output quantity] when [input variable] is 6". Finally, scan the choices and eliminate any that: (1) turn the function value into a rate ("per hour"), (2) change units (like turning milligrams into percent), or (3) mix up the numbers or the time described; the remaining option should mirror your plain-language translation.
Hints
Clarify input and output
Ask yourself: In the function , what does the input represent, and what does the output represent? Are they times, amounts, rates, or percents?
Focus on A(6) in context
means you substitute into the function. What moment in time does describe, and what quantity is the function giving you at that moment?
Use the given approximation
The problem tells you that is approximately . How can you describe, in words, what it means for the function's output to be about 14 at ?
Eliminate rate and percent interpretations
Check whether any choice is incorrectly turning the function's output into a "per hour" rate or a percent of the original amount. itself is just a single amount at one time.
Desmos Guide
Enter the function
In Desmos, type the expression y = 40*(0.84)^x to graph the amount of tracer (y, in milligrams) as a function of time (x, in hours after noon).
Evaluate the function at 6 hours
In a new line, type 40*(0.84)^6 and let Desmos calculate the value. The result is the amount of tracer, in milligrams, that remains in the body 6 hours after noon; use this value to match the correct verbal description.
Step-by-step Explanation
Identify what the function represents
The function is
- The input is time in hours after noon.
- The output is the amount of tracer in milligrams remaining in the body at that time.
So always represents an amount (in mg), not a rate and not a percent.
Interpret the meaning of A(6)
The expression means:
- Plug in hours after noon.
- The value is the amount of tracer (in mg) 6 hours after noon.
We are told is approximately equal to , so the amount at that time is about 14 milligrams.
(Optional) Check the calculation numerically
You can verify the approximation by evaluating the function at :
If you compute with a calculator and multiply by 40, you get a value close to 14. This confirms that the model predicts an amount of about 14 milligrams at hours.
Match the interpretation to the answer choice
From the context:
- The time is 6 hours after noon.
- The amount of tracer in the body at that time is about 14 milligrams.
The only choice that says exactly this is:
About 14 milligrams of the tracer remain in the body 6 hours after noon.