The question is asking to project the value of $f(15)$, where $f(t)$ represents the US debt (in billions of dollars) at time $t$ years after 2005, using the data provided in the table. It asks to assume that:

$\frac{f(15) - f(7)}{15 - 7} = \frac{f(7) - f(2)}{7 - 2}$

Step-by-Step Solution:

1. Extract values from the table:

   • $f(7) = 15000$ (from the table for $t = 7$)
   • $f(2) = 9000$ (from the table for $t = 2$)

2. Use the second part of the equation: $\frac{f(7) - f(2)}{7 - 2} = \frac{15000 - 9000}{7 - 2} = \frac{6000}{5} = 1200$ Therefore, the slope of the linear change between $t = 2$ and $t = 7$ is 1200.

3. Apply this slope to estimate $f(15)$: $\frac{f(15) - f(7)}{15 - 7} = 1200$ This simplifies to: $f(15) - 15000 = 1200 \times (15 - 7)$ $f(15) - 15000 = 1200 \times 8 = 9600$ $f(15) = 15000 + 9600 = 24600$

Final Answer:

The projected value of $f(15)$ is 24600 billion dollars.

Would you like further details on any step or have additional questions?

Here are five related questions:

1. What does the slope represent in the context of this debt data?
2. How would the projection change if the slope between different years was used?
3. What assumptions are made when using linear projections like this?
4. Can you calculate $f(20)$ using the same method?
5. How accurate are linear models for predicting financial data over long periods?

Tip: Linear projection assumes a constant rate of change, which may not always be accurate for real-world data, especially over extended periods.