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Two art museums are hosting new long-term exhibits. The number of daily visitors attending each exhibit is modeled by functions [tex]$v$[/tex] and [tex]$s$[/tex], where [tex]$n$[/tex] is the number of days since the exhibits opened.

Visual Arts Exhibit
[tex]\[ v(n) = \frac{50}{n} + 120 \][/tex]

Sculpture Exhibit
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
n & 10 & 50 & 100 & 120 & 200 \\
\hline
s(n) & 210 & 154 & 147 & 145 & 143 \\
\hline
\end{array}
\][/tex]

Which statement accurately describes this situation?
A. As the number of days increases, the number of daily visitors at the visual arts exhibit levels off to a lower amount than the number of daily visitors at the sculpture exhibit.
B. As the number of days increases, the number of daily visitors at both exhibits decreases to zero.
C. As the number of days increases, the number of daily visitors at the visual arts exhibit levels off to a higher amount than the number of daily visitors at the sculpture exhibit.
D. As the number of days increases, the number of daily visitors at the visual arts exhibit levels off to the same amount as the number of daily visitors at the sculpture exhibit.

Sagot :

GinnyAnsweridn
To determine which statement accurately describes the situation, let's analyze the behavior of both functions [tex]\( v(n) \)[/tex] and [tex]\( s(n) \)[/tex] as [tex]\( n \)[/tex] (the number of days since the exhibits opened) increases.

### Visual Arts Exhibit
The function for the number of daily visitors at the Visual Arts Exhibit is given by:
[tex]\[ v(n) = \frac{50}{a} + 120 \][/tex]

Notice that in this equation, the term [tex]\(\frac{50}{a}\)[/tex] does not directly depend on [tex]\( n \)[/tex]. Given that [tex]\( a \)[/tex] might be related to [tex]\( n \)[/tex] in practice, let's consider [tex]\( a = n \)[/tex] for understanding the behavior as [tex]\( n \)[/tex] increases.

When [tex]\( a = n \)[/tex]:
[tex]\[ v(n) = \frac{50}{n} + 120 \][/tex]

As [tex]\( n \)[/tex] increases, the term [tex]\(\frac{50}{n}\)[/tex] becomes very small because the denominator gets larger. Thus, [tex]\( \frac{50}{n} \)[/tex] approaches 0. Therefore, [tex]\( v(n) \)[/tex] approaches:
[tex]\[ v(n) \approx 120 \][/tex]

### Sculpture Exhibit
We are given the following values:

[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline n & 10 & 50 & 100 & 120 & 200 \\ \hline s(n) & 210 & 154 & 147 & 145 & 143 \\ \hline \end{array} \][/tex]

By examining the values in the table, we can observe that as [tex]\( n \)[/tex] increases, [tex]\( s(n) \)[/tex] decreases. It seems to decrease progressively slower and appears to level off. For large values of [tex]\( n \)[/tex], [tex]\( s(n) \)[/tex] seems to approach a value around 143.

### Conclusion
Comparing the behaviors of both functions as [tex]\( n \)[/tex] increases:
- [tex]\( v(n) \)[/tex] levels off to 120.
- [tex]\( s(n) \)[/tex] levels off to around 143.

Thus, as the number of days increases, the number of daily visitors at the Visual Arts Exhibit levels off to 120, which is a lower amount compared to the daily visitors at the Sculpture Exhibit leveling off to around 143.

Therefore, the correct answer is:
A. As the number of days increases, the number of daily visitors at the visual arts exhibit levels off to a lower amount than the number of daily visitors at the sculpture exhibit.
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