At most, Alana can spend $40 on carnival tickets. Ride tickets cost $4 each, and food tickets cost $2 each. Alana buys at least 16 tickets. The system of inequalities represents the number of ride tickets, $r$, and the number of food tickets, $f$, she buys.
[tex]
\begin{array}{c}
r+f \geq 16 \\
4 r+2 f \leq 40
\end{array}
[/tex]
What is the maximum number of ride tickets she can buy?
A. 4
B. 6
C. 10
D. 12
Answer (1)
Express f in terms of r from both inequalities: f g e q 16 − r and f ≤ 20 − 2 r .
Combine the inequalities to find the range for r : 16 − r ≤ 20 − 2 r .
Solve for r : r ≤ 4 .
The maximum number of ride tickets Alana can buy is 4 .
Explanation
Problem Analysis Let's analyze the problem. We are given two inequalities:
r + f ≥ 16 4 r + 2 f ≤ 40
where r is the number of ride tickets and f is the number of food tickets. We want to find the maximum number of ride tickets, r , that Alana can buy.
Simplifying the Inequalities First, let's simplify the second inequality by dividing both sides by 2:
2 r + f ≤ 20
Now we have the system:
r + f ≥ 16 2 r + f ≤ 20
Isolating f in the First Inequality From the first inequality, we can express f in terms of r :
f ≥ 16 − r
Isolating f in the Second Inequality From the second inequality, we can also express f in terms of r :
f ≤ 20 − 2 r
Combining the Inequalities Now we can combine these two inequalities to find the possible values of r :
16 − r ≤ f ≤ 20 − 2 r
This implies that:
16 − r ≤ 20 − 2 r
Solving for r Now, let's solve for r :
2 r − r ≤ 20 − 16 r ≤ 4
Checking the Solution Since r must be an integer, the maximum possible value for r is 4. Let's check if this value is feasible. If r = 4 , then:
f ≥ 16 − 4 = 12 f ≤ 20 − 2 ( 4 ) = 20 − 8 = 12
So, f = 12 . This means Alana can buy 4 ride tickets and 12 food tickets.
Final Answer Therefore, the maximum number of ride tickets Alana can buy is 4.
Examples
Understanding systems of inequalities can help in resource allocation problems. For instance, a farmer has a limited budget and land to plant two crops. Each crop has different costs and yields different profits. By setting up a system of inequalities, the farmer can determine the optimal amount of each crop to plant to maximize profit while staying within budget and land constraints. This ensures efficient use of resources and maximizes potential earnings.
