The members of the drama club have 100 tickets to sell to the school play. Students pay $5 per ticket, and nonstudents pay $10 per ticket. The drama club needs to collect at least $800 in total ticket sales. The system of inequalities represents the number of student tickets, $s$, and the number of nonstudent tickets, $n$, the members must sell.
What is the maximum number of student tickets that can be sold if the drama club meets its sales goal?
[tex]
\begin{array}{r}
s+n \leq 100 \\
5 s+10 n \geq 800
\end{array}
[/tex]
A. 40
B. 60
C. 80
D. 100
Answer (2)
Express n in terms of s from the first inequality: $n
Substitute this expression for n into the second inequality: $5s + 10(100 - s)
Simplify and solve for s : $s
The maximum number of student tickets that can be sold is 40 .
Explanation
Problem Analysis Let's analyze the problem. We have two inequalities:
s + n 5s + 10n
We want to find the maximum value of s that satisfies both inequalities.
Express n in terms of s First, let's express n in terms of s using the first inequality:
s + n n
So, n must be less than or equal to 100 − s .
Solve for s Now, substitute this expression for n into the second inequality:
5 s + 10 n 5s + 10(100 - s)
Simplify the inequality:
5 s + 1000 − 10 s -5s + 1000 − 5 s s
So, s must be less than or equal to 40.
Find corresponding n To maximize s , let's consider the case when s = 40 . Substitute s = 40 into the first inequality to find the corresponding value of n :
40 + n n
Substitute s = 40 into the second inequality to check if the sales goal is met:
5 ( 40 ) + 10 n 200 + 10n 10 n n
Since $n
Final Answer Since $n
Therefore, the maximum number of student tickets that can be sold is 40.
Examples
Understanding systems of inequalities can help in resource allocation. For example, a store owner has a budget and limited shelf space. They need to decide how many of each item to stock to maximize profit while staying within budget and space constraints. The inequalities would represent the budget constraint and the space constraint, and the variables would be the quantities of each item. Solving the system of inequalities would help the store owner determine the optimal quantities of each item to stock.
The maximum number of student tickets that can be sold while meeting the sales goal of at least $800 is 40. This is found by solving the inequalities that represent the constraints on ticket sales. Therefore, the answer is A. 40.
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