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A construction crew is lengthening a road. Let [tex]\( L \)[/tex] be the total length of the road (in miles). Let [tex]\( D \)[/tex] be the number of days the crew has worked. Suppose that [tex]\( L = 2D + 200 \)[/tex] gives [tex]\( L \)[/tex] as a function of [tex]\( D \)[/tex]. The crew can work for at most 60 days.

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}
number of days the crew has worked \\
length of the road (in miles)
\end{tabular} & (Choose one) \\
\hline Range: & \begin{tabular}{l}
length of the road (in miles) \\
number of days the crew has worked
\end{tabular} & (Choose one) \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
Let's consider the given information step-by-step:

1. Understanding the function:
We have the function [tex]\( L = 2D + 200 \)[/tex], which describes the length of the road [tex]\( L \)[/tex] as a function of the number of days [tex]\( D \)[/tex] the crew has worked. This tells us that for each value of [tex]\( D \)[/tex], we can determine a corresponding value of [tex]\( L \)[/tex].

2. Identifying the domain:
The domain of a function is the set of all possible input values (in this case, values of [tex]\( D \)[/tex]) that the function can take. According to the problem, the crew can work for at most 60 days. Therefore, the number of days [tex]\( D \)[/tex] can range from 0 to 60.
So, the correct description of the domain is "number of days the crew has worked" and the set of values is [tex]\( D ∈ [0, 60] \)[/tex].

3. Identifying the range:
The range of a function is the set of all possible output values (in this case, values of [tex]\( L \)[/tex]) that the function can produce. Given that [tex]\( D \)[/tex] ranges from 0 to 60, we can find the corresponding range for [tex]\( L \)[/tex]:
- When [tex]\( D = 0 \)[/tex]:
[tex]\( L = 2(0) + 200 = 200 \)[/tex]
- When [tex]\( D = 60 \)[/tex]:
[tex]\( L = 2(60) + 200 = 320 \)[/tex]
Thus, the length of the road [tex]\( L \)[/tex] will range from 200 miles to 320 miles.
So, the correct description of the range is "length of the road (in miles)" and the set of values is [tex]\( L ∈ [200, 320] \)[/tex].

4. Filling in the table correctly:

[tex]\[ \begin{tabular}{|l|l|l|} \hline & Description of Values & Set of Values \\ \hline Domain: & \begin{tabular}{l} O length of the road (in miles) \\ ✔ number of days the crew has worked \\ \end{tabular} & \( D ∈ [0, 60] \) \\ \hline Range: & \begin{tabular}{l} ✔ length of the road (in miles) \\ O number of days the crew has worked \\ \end{tabular} & \( L ∈ [200, 320] \) \\ \hline \end{tabular} \][/tex]

In conclusion:

- The domain represents the number of days the crew has worked and is [tex]\( D ∈ [0, 60] \)[/tex].
- The range represents the length of the road (in miles) and is [tex]\( L ∈ [200, 320] \)[/tex].
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