The given table represents data that is best modeled by an exponential function. To determine which of the provided equations fits the data, let's analyze the options step by step.

Steps to Solve

1. General Exponential Form:
   An exponential model takes the form:
   $y = a \cdot b^x$
   where $a$ is the initial value (when $x = 0$) and $b$ is the base, representing the growth/decay factor.

2. Data Check:
   From the table:

   • At $x = 1$, $f(x) = 850$
   • At $x = 15$, $f(x) = 720$
   • At $x = 35$, $f(x) = 610$
   • At $x = 55$, $f(x) = 530$
   • At $x = 75$, $f(x) = 440$

   This indicates a decay model because the values decrease as $x$ increases.

3. Compare the Equations:
   Evaluate each provided model to see which closely fits this decay trend:

   • Option 1: $y = 837.782 \cdot (0.991)^x$
   • Option 2: $y = 850 \cdot (0.983)^x$
   • Option 3: $y = 0.001 \cdot (0.156)^x$
   • Option 4: $y = 7.218 \cdot (0.002)^x$

   Both Option 3 and Option 4 are unlikely because their $b$ values (0.156 and 0.002) indicate an extremely rapid decay, which does not match the given gradual decrease.

4. Refine Between Option 1 and 2:

   • The initial value in Option 1 ($a = 837.782$) is close to 850 but slightly adjusted due to fitting the model.
   • The decay factor in Option 1 ($b = 0.991$) indicates a slower decay, while in Option 2, $b = 0.983$, indicating faster decay.

   To fit the data points better, Option 1 appears most plausible.

Final Answer:

The best equation is:
$y = 837.782 \cdot (0.991)^x$

Would you like further details on the calculation or another breakdown? Here are related questions to expand understanding:

1. How can you derive $a$ and $b$ for exponential models manually?
2. How does exponential decay differ from linear decay?
3. Can you confirm this model using logarithmic transformations?
4. What other regression tools can be used to verify $a$ and $b$?
5. How would changing $b$ slightly (e.g., from 0.991 to 0.95) impact the graph?

Tip: Always check the initial value ($a$) and the base ($b$) to ensure the model matches the general trend of your data!