The following table shows the percentage P = P(d) of the American food dollar that was spent on eating away from home (at restaurants, for example) as a function of the date d.
d = Year	P = Percent spent
away from home
1969	25%
1989	30%
2009	34%
Estimate P(1999).

To estimate \( P(1999) \), we can assume that the relationship between the percentage spent away from home \( P(d) \) and the year \( d \) is linear between the given points. In other words, we can apply linear interpolation between the years 1989 and 2009 to estimate the value for 1999.

Here’s the step-by-step process:

1. We have two points:  
   - \( (1989, 30\%) \)  
   - \( (2009, 34\%) \)
   
2. The slope \( m \) of the line between these two points is:
   \[
   m = \frac{P(2009) - P(1989)}{2009 - 1989} = \frac{34\% - 30\%}{2009 - 1989} = \frac{4}{20} = 0.2 \% \text{ per year}
   \]

3. Using the slope, we can estimate \( P(1999) \). First, find the difference in years between 1989 and 1999, which is \( 1999 - 1989 = 10 \) years. Then, apply the slope to calculate the change in percentage:
   \[
   P(1999) = P(1989) + (m \times 10) = 30\% + (0.2 \times 10) = 30\% + 2\% = 32\%
   \]

Thus, \( P(1999) \) is estimated to be **32%**.

Would you like more details on the linear interpolation method?

Here are 5 related questions to expand your understanding:
1. How do you perform linear interpolation for other data points?
2. What is the difference between linear interpolation and extrapolation?
3. How do you determine if a data set is linear or not?
4. How can you model non-linear data?
5. What are the limitations of using linear interpolation?

**Tip:** Linear interpolation works best for estimates between two known data points. For points outside the range, extrapolation should be used, which can be less reliable.

The following table shows the percentage P = P(d) of the American food dollar that was spent on eating away from home (at restaurants, for example) as a function of the date d.

 d = Year   P = Percent spent away from home
 1969   25%
 1989   30%
 2009   34%

Estimate P(1999).

Learn how to estimate the percentage of the American food dollar spent away from home in 1999 using linear interpolation. This problem involves calculating the slope between two data points from 1989 and 2009 and applying it to estimate the value for 1999. Suitable for students in Grades 9-12.

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