To solve the problem of determining the maximum number of ride tickets Alana can buy given the constraints, let's look at the inequalities provided and analyze the feasible region step-by-step.
We have the following system of inequalities:
1. [tex]\( r + f \geq 16 \)[/tex]
2. [tex]\( 4r + 2f \leq 40 \)[/tex]
### Step-by-Step Solution:
1. Simplify the second inequality:
[tex]\[
4r + 2f \leq 40
\][/tex]
Divide the entire inequality by 2:
[tex]\[
2r + f \leq 20
\][/tex]
2. Graphical interpretation:
To find the feasible region, we can interpret these inequalities graphically, but since we can evaluate integer values, we inspect possible solutions algebraically.
3. Inspect possible values:
We need to find the maximum integer value of [tex]\( r \)[/tex] such that both inequalities are satisfied.
Let's express [tex]\( f \)[/tex] in terms of [tex]\( r \)[/tex] from the simplified second inequality:
[tex]\[
f \leq 20 - 2r
\][/tex]
And from the first inequality:
[tex]\[
f \geq 16 - r
\][/tex]
For each value of [tex]\( r \)[/tex], [tex]\( f \)[/tex] must satisfy:
[tex]\[
16 - r \leq f \leq 20 - 2r
\][/tex]
4. Determine feasible [tex]\( r \)[/tex] values:
[tex]\[
16 - r \leq 20 - 2r
\][/tex]
Solve this inequality:
[tex]\[
16 - r \leq 20 - 2r
\][/tex]
Add [tex]\( 2r \)[/tex] to both sides:
[tex]\[
r \leq 20 - 16
\][/tex]
[tex]\[
r \leq 4
\][/tex]
5. Check boundary [tex]\( r = 4 \)[/tex]:
If [tex]\( r = 4 \)[/tex]:
[tex]\[
f \leq 20 - 2 \times 4
\][/tex]
[tex]\[
f \leq 12
\][/tex]
[tex]\[
f \geq 16 - 4
\][/tex]
[tex]\[
f \geq 12
\][/tex]
So, when [tex]\( r = 4 \)[/tex], [tex]\( f = 12 \)[/tex] is feasible and satisfies both inequalities:
- [tex]\( 4 + 12 = 16 \)[/tex] (satisfies [tex]\( r + f \geq 16 \)[/tex])
- [tex]\( 4 \times 4 + 2 \times 12 = 16 + 24 = 40 \)[/tex] (satisfies [tex]\( 4r + 2f \leq 40 \)[/tex])
6. Conclusion:
Therefore, the maximum number of ride tickets [tex]\( r \)[/tex] that Alana can buy, under the given constraints, is:
[tex]\[ \boxed{4} \][/tex]
