Question

A garden designer is working on a project where they need to produce a certain number of flower beds and garden paths. The inequality 25x+15y≥200 represents the production goal, where x is the number of flower beds and y is the number of garden paths. The graph below shows various points on a coordinate plane, with each point representing a possible combination of flower beds and garden paths.Determine which of these points are viable combinations that meet the designer's production goal,
(2,10)
None of the points are a viable solution
(3,8)
(4,6)
(6,2)

Answer 1

To determine which points meet the designer's production goal represented by the inequality \( 25x + 15y \geq 200 \), we will substitute each point into the inequality and check if it satisfies the condition.

1. **Point (2, 10)**:
\[
25(2) + 15(10) = 50 + 150 = 200
\]
Since \( 200 \geq 200 \) is true, (2, 10) is a viable solution.

2. **Point (3, 8)**:
\[
25(3) + 15(8) = 75 + 120 = 195
\]
Since \( 195 \geq 200 \) is false, (3, 8) is not a viable solution.

3. **Point (4, 6)**:
\[
25(4) + 15(6) = 100 + 90 = 190
\]
Since \( 190 \geq 200 \) is false, (4, 6) is not a viable solution.

4. **Point (6, 2)**:
\[
25(6) + 15(2) = 150 + 30 = 180
\]
Since \( 180 \geq 200 \) is false, (6, 2) is not a viable solution.

After checking all the points, the only viable combination that meets the designer's production goal is **(2, 10)**.

Thus, the answer is:
- Viable points: (2, 10)
- Non-viable points: (3, 8), (4, 6), (6, 2)