The problem describes the exponential decay of a radioactive substance and asks for the function that models the amount left at the end of $x$ days. Let’s solve it step by step:

Problem Breakdown:

1. Initial amount: $100$ grams.
2. Remaining amount after 3 days: $72.9$ grams.
3. The model is exponential, of the form: $f(x) = A \cdot r^x$ Where:
   • $A$ is the initial amount ($100$ grams).
   • $r$ is the decay factor per day.
   • $x$ is the number of days.

Step 1: Substitute known values

From the given information, at $x = 3$, $f(3) = 72.9$. Substituting into the model: $72.9 = 100 \cdot r^3$

Step 2: Solve for $r$ (the decay factor)

Divide both sides by $100$: $r^3 = \frac{72.9}{100} = 0.729$ Now take the cube root: $r = \sqrt[3]{0.729} = 0.9$

Step 3: Write the equation

The decay factor is $r = 0.9$, and the initial amount is $100$. The function is: $f(x) = 100 \cdot (0.9)^x$

Final Answer:

The correct option is D: $f(x) = 100 \cdot (0.9)^x$.

Would you like a detailed explanation of any step? Let me know!

Related Questions:

1. How is the decay factor $r$ related to the half-life of a substance?
2. What changes if the initial amount is doubled but the decay factor remains the same?
3. Can you derive a general formula to find $r$ given any initial amount and time?
4. How does changing $r$ affect the graph of the exponential function?
5. How is the concept of exponential decay applied in real-world scenarios like medicine or physics?

Tip: When solving exponential decay problems, make sure to isolate the decay factor step-by-step by working with logarithms or roots as needed.