Answer :
To solve this problem, we want to set up a system of inequalities that will help us determine the number of football players ([tex]\(f\)[/tex]) and volleyball players ([tex]\(v\)[/tex]) that can talk to the student body during the pep rally, given certain restrictions.
1. Determine the constraints:
- Player Constraint: The total number of players talking (both football and volleyball) should not exceed 11. This gives us the inequality:
[tex]\[
f + v \leq 11
\][/tex]
- Time Constraint: The total speaking time for all players should not exceed 242 seconds. Football players speak for 17 seconds each, and volleyball players speak for 28 seconds each. Therefore, the inequality representing the total time constraint is:
[tex]\[
17f + 28v \leq 242
\][/tex]
2. Formulate the system of inequalities:
Using the constraints identified:
- The first inequality, [tex]\(f + v \leq 11\)[/tex], ensures the sum of the football and volleyball players talking is at most 11.
- The second inequality, [tex]\(17f + 28v \leq 242\)[/tex], ensures the total speaking time for all players is at most 242 seconds.
3. Reviewing the options:
Given the formulated inequalities, we need to select the correct option that reflects these constraints accurately:
- Option A:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v < 242\)[/tex]
The second inequality uses "<" instead of "≤", which does not match our conditions since the total time can be equal to 242 seconds.
- Option B:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v \leq 242\)[/tex]
This option correctly uses the "≤" sign in both inequalities, therefore, it aligns with our formulated constraints.
- Option C:
- [tex]\(f + v < 11\)[/tex]
- [tex]\(17v + 28f \leq 242\)[/tex]
This option has both a different player constraint and a switched coefficient for [tex]\(f\)[/tex] and [tex]\(v\)[/tex] in the second inequality, which does not correspond to our inequalities.
Therefore, the correct system of inequalities to solve the problem is given by Option B:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v \leq 242\)[/tex]
1. Determine the constraints:
- Player Constraint: The total number of players talking (both football and volleyball) should not exceed 11. This gives us the inequality:
[tex]\[
f + v \leq 11
\][/tex]
- Time Constraint: The total speaking time for all players should not exceed 242 seconds. Football players speak for 17 seconds each, and volleyball players speak for 28 seconds each. Therefore, the inequality representing the total time constraint is:
[tex]\[
17f + 28v \leq 242
\][/tex]
2. Formulate the system of inequalities:
Using the constraints identified:
- The first inequality, [tex]\(f + v \leq 11\)[/tex], ensures the sum of the football and volleyball players talking is at most 11.
- The second inequality, [tex]\(17f + 28v \leq 242\)[/tex], ensures the total speaking time for all players is at most 242 seconds.
3. Reviewing the options:
Given the formulated inequalities, we need to select the correct option that reflects these constraints accurately:
- Option A:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v < 242\)[/tex]
The second inequality uses "<" instead of "≤", which does not match our conditions since the total time can be equal to 242 seconds.
- Option B:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v \leq 242\)[/tex]
This option correctly uses the "≤" sign in both inequalities, therefore, it aligns with our formulated constraints.
- Option C:
- [tex]\(f + v < 11\)[/tex]
- [tex]\(17v + 28f \leq 242\)[/tex]
This option has both a different player constraint and a switched coefficient for [tex]\(f\)[/tex] and [tex]\(v\)[/tex] in the second inequality, which does not correspond to our inequalities.
Therefore, the correct system of inequalities to solve the problem is given by Option B:
- [tex]\(f + v \leq 11\)[/tex]
- [tex]\(17f + 28v \leq 242\)[/tex]
