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An amusement park is installing a new roller coaster. The park intends to charge [tex]$5$[/tex] per adult and [tex]$3$[/tex] per child for each ride. It hopes to earn back more than the [tex]$750,000$[/tex] cost of construction in four years. With the best of weather, the park can provide 100,000 adult rides and 75,000 child rides in one season.

Let [tex]$x$[/tex] be the number of adult rides in 4 years and [tex]$y$[/tex] be the number of child rides in 4 years. Which system of inequalities best represents this situation?

A.
[tex]\[
\begin{array}{l}
5x + 3y \leq 750,000 \\
x \leq 100,000 \\
y \leq 75,000 \\
\end{array}
\][/tex]

B.
[tex]\[
\begin{array}{l}
5x + 3y \geq 750,000 \\
x \leq 100,000 \\
y \leq 75,000 \\
\end{array}
\][/tex]

C.
[tex]\[
\begin{array}{l}
5x + 3y \geq 750,000 \\
x \leq 400,000 \\
y \leq 300,000 \\
\end{array}
\][/tex]

D.
[tex]\[
\begin{array}{l}
5x + 3y \leq 750,000 \\
x \leq 400,000 \\
y \leq 300,000 \\
\end{array}
\][/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to represent the conditions given in the question with a system of inequalities.

1. The park charges [tex]$5 per adult ride and $[/tex]3 per child ride. Thus, the total revenue from adult rides and child rides in 4 years can be expressed as [tex]\(5x + 3y\)[/tex], where [tex]\(x\)[/tex] is the number of adult rides and [tex]\(y\)[/tex] is the number of child rides over the 4 years.

2. The park wants to earn back more than $750,000 for the construction cost. Therefore, the inequality representing this condition is:
[tex]\[ 5x + 3y \geq 750,000 \][/tex]

3. The best weather constraints are that the park can provide at most 100,000 adult rides and 75,000 child rides each season. Over 4 seasons, this translates to:
[tex]\[ x \leq 4 \times 100,000 = 400,000 \][/tex]
[tex]\[ y \leq 4 \times 75,000 = 300,000 \][/tex]

So, the correct system of inequalities that represents this situation is:
[tex]\[ \begin{array}{c} 5x + 3y \geq 750,000 \\ x \leq 400,000 \\ y \leq 300,000 \\ \end{array} \][/tex]

Therefore, the correct answer is:
[tex]\[ \begin{array}{c} 5x + 3y \geq 750,000 \quad x \leq 400,000 \\ y \leq 300,000 \end{array} \][/tex]
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