The function S(t)=1.084t+17.392
 models the amount of money, S
, in billions of dollars, the United States government spent on science, space, and technology for the time period 1999-2009, where t
 is the number of years since 1999.

 

What are the constraints for the variable t
 using set-builder notation?

The function \( S(t) = 1.084t + 17.392 \) models the amount of money spent on science, space, and technology by the U.S. government for the years 1999 to 2009, where \( t \) is the number of years since 1999.

Since \( t \) represents the number of years since 1999, we can determine the constraints on \( t \) by considering the time period from 1999 to 2009:

- For the year 1999, \( t = 0 \).
- For the year 2009, \( t = 2009 - 1999 = 10 \).

Thus, the value of \( t \) ranges from 0 to 10, inclusive.

### Set-builder notation for the constraints on \( t \):

\[
\{ t \in \mathbb{R} \mid 0 \leq t \leq 10 \}
\]

This means that \( t \) is a real number, and it lies between 0 and 10, inclusive.

Would you like further details or have any questions? Here are some related questions you might explore:

1. What is the interpretation of the slope \( 1.084 \) in the context of this problem?
2. How much money was spent in the year 2005 according to the model?
3. What does the value 17.392 represent in the context of the model?
4. How would the model change if we wanted to extend it beyond 2009?
5. What is the predicted amount spent in the year 2010 if the model were extended?

**Tip:** When interpreting real-world models, always identify what each variable and constant represents to better understand the context.

The function S(t) = 1.084t + 17.392 models the amount of money, S, in billions of dollars, the United States government spent on science, space, and technology for the time period 1999-2009, where t is the number of years since 1999. What are the constraints for the variable t using set-builder notation?

Explore how to determine the constraints for the variable t in the linear function S(t) = 1.084t + 17.392, which models the amount of money spent by the U.S. government on science, space, and technology from 1999 to 2009. Learn how to express these constraints using set-builder notation. Suitable for students in Grades 9-11.

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