Question

1. you are planning a school field trip to a local amusement park. the park charges an admission fee of $25 per student and $30 per adult. you want to ensure that the total amount collected from the group is at least $500. additionally, there are at least 5 students attending the trip, and the number of adults cannot exceed 20. part a: choose the system of inequalities that best represents the real - world situation described. a. \\(\

$$\begin{cases}25x + 30y\\leq500\\\\x\\geq5\\\\y\\leq20\\end{cases}$$

\\) b. \\(\

$$\begin{cases}25x + 30y\\geq500\\\\x\\leq5\\\\y\\leq20\\end{cases}$$

\\) c. \\(\

$$\begin{cases}25x + 30y\\geq500\\\\x\\geq5\\\\y\\leq20\\end{cases}$$

\\) d. \\(\

$$\begin{cases}25x + 30y\\leq500\\\\x\\leq5\\\\y\\leq20\\end{cases}$$

\\) part b: determine whether the following combinations of students and adults are viable or non - viable. | | viable | non - viable | | --- | --- | --- | | 10 students and 10 adults | \\(\bigcirc\\) | \\(\bigcirc\\) | | 20 students and 5 adults | \\(\bigcirc\\) | \\(\bigcirc\\) | | 5 students and 15 adults | \\(\bigcirc\\) | \\(\bigcirc\\) | | 15 students and 3 adults | \\(\bigcirc\\) | \\(\bigcirc\\) | | 3 students and 15 adults | \\(\bigcirc\\) | \\(\bigcirc\\) |

Explanation:

Step1: Analyze total collection condition

The total amount collected ($25x + 30y$, where $x$ = number of students, $y$ = number of adults) must be at least $500$, so $25x + 30y \geq 500$.

Step2: Analyze student count condition

At least 5 students attend, so $x \geq 5$.

Step3: Analyze adult count condition

Adults cannot exceed 20, so $y \leq 20$.

Step4: Verify Part B combinations

For each pair, check if it satisfies all 3 inequalities:

1. 10 students, 10 adults: $25(10)+30(10)=550\geq500$, $10\geq5$, $10\leq20$ → Viable
2. 20 students, 5 adults: $25(20)+30(5)=650\geq500$, $20\geq5$, $5\leq20$ → Viable
3. 5 students, 15 adults: $25(5)+30(15)=575\geq500$, $5\geq5$, $15\leq20$ → Viable
4. 15 students, 3 adults: $25(15)+30(3)=465<500$ → Non-Viable
5. 3 students, 15 adults: $3<5$ → Non-Viable

Answer:

Part A:

C.

$$\begin{cases} 25x + 30y \geq 500 \\ x \geq 5 \\ y \leq 20 \end{cases}$$

Part B:

1. 10 students and 10 adults: Viable
2. 20 students and 5 adults: Viable
3. 5 students and 15 adults: Viable
4. 15 students and 3 adults: Non-Viable
5. 3 students and 15 adults: Non-Viable