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Document: Exam: 02.11 Markets and You

Graphs and Functions

Domain and range of a linear function that models a real-world situation.

The owners of a recreation area are filling a small pond with water. Let [tex]$W$[/tex] be the total amount of water in the pond (in liters). Let [tex]$T$[/tex] be the total number of minutes that water has been added. Suppose that [tex]$W = 25T + 700$[/tex] gives [tex][tex]$W$[/tex][/tex] as a function of [tex]$T$[/tex] during the next 80 minutes.

Identify the correct description of the values in both the domain and range of the function. Then, for each, choose the most appropriate set of values.

\begin{tabular}{|l|l|l|}
\hline & Description of Values & Set of Values \\
\hline Domain: & \begin{tabular}{l}
amount of water in the pond (in liters) \\
number of minutes water has been added
\end{tabular} & (Choose one) \\
\hline Range: & \begin{tabular}{l}
amount of water in the pond (in liters) \\
number of minutes water has been added
\end{tabular} & (Choose one) \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
Let's analyze the given problem step-by-step to determine the domain and range of the function [tex]\( W = 25T + 700 \)[/tex], where [tex]\( W \)[/tex] represents the total amount of water in liters in the pond and [tex]\( T \)[/tex] represents the time in minutes that water has been added.

### Step 1: Identify the Domain
The domain of a function is the set of all possible input values. In this problem:
- [tex]\( T \)[/tex] (time in minutes) is the input value.

The time [tex]\( T \)[/tex] can vary from the moment the water starts to be added until the end of the 80 minutes.

Description of Values:
- Domain: Number of minutes water has been added.

Set of Values:
- The time interval [tex]\( T \)[/tex] ranges from 0 to 80 minutes, inclusive.
- Therefore, the domain is [tex]\( (0, 80) \)[/tex].

### Step 2: Identify the Range
The range of a function is the set of all possible output values. In this problem:
- [tex]\( W \)[/tex] (amount of water in liters) is the output value.

To determine the range, we need to evaluate the amount of water in the pond at the start and the end of the time interval.

- When [tex]\( T = 0 \)[/tex] minutes, [tex]\( W = 25(0) + 700 = 700 \)[/tex] liters.
- When [tex]\( T = 80 \)[/tex] minutes, [tex]\( W = 25(80) + 700 = 2000 + 700 = 2700 \)[/tex] liters.

Description of Values:
- Range: Amount of water in the pond (in liters).

Set of Values:
- The amount of water ranges from 700 liters to 2700 liters.
- Therefore, the range is [tex]\( (700, 2700) \)[/tex].

### Conclusion
Putting it all together:
- Domain:
- Description of Values: Number of minutes water has been added.
- Set of Values: (0, 80)
- Range:
- Description of Values: Amount of water in the pond (in liters).
- Set of Values: (700, 2700)

This analysis clearly outlines the domain and range for the function [tex]\( W = 25T + 700 \)[/tex] that models the amount of water being added to the pond over the next 80 minutes.
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