Question

1. the function ( s(t) = 3.4t + 8.62 ) models the amount of raider bucks mr. lepage will give from the years 2022 - 2031, where ( t ) is the number of years from 2022.

a. write the constraints for the variable ( t ) using set - builder notation.
constraints
b. determine the value of ( s(7) ).
c. interpret the value of ( s(7) ) in context.
d. name the key feature that is found when solving ( s(0) = 8.62 ). then, label the key feature on the graph of ( s(t) = 3.4t + 8.62 ).
key feature

Explanation:

Part a:

Step1: Identify time range

The years are from 2022 to 2031, so \( t \) (years from 2022) ranges from \( 0 \) (2022) to \( 9 \) (2031, since \( 2031 - 2022 = 9 \)).

Step2: Write set - builder notation

Using set - builder notation, the constraint for \( t \) is \( \{t\in\mathbb{R}\mid0\leq t\leq9\} \), where \( \mathbb{R} \) represents the set of real numbers (since \( t \) is a number of years, it can be considered as a real number in this context, and we restrict it to the interval from 0 to 9 inclusive).

Step1: Substitute \( t = 7 \) into the function

We are given the function \( S(t)=3.4t + 8.62 \). To find \( S(7) \), we substitute \( t = 7 \) into the function.

Step2: Calculate the value

\( S(7)=3.4\times7 + 8.62 \). First, calculate \( 3.4\times7=23.8 \). Then, add \( 8.62 \) to \( 23.8 \): \( 23.8+8.62 = 32.42 \).

The function \( S(t) \) models the amount of raider bucks Mr. Lepage will give. When \( t = 7 \), it means 7 years from 2022, which is the year 2022 + 7=2029. The value \( S(7) = 32.42 \) means that in the year 2029, the amount of raider bucks Mr. Lepage will give is 32.42 (the unit of raider bucks is not specified in the problem, but we can interpret the value in the context of the function's purpose).

Answer:

\( \{t\in\mathbb{R}\mid0\leq t\leq9\} \)

Part b: