Question

Anton wants to plant a small orchard of apple trees and cherry trees in his back yard. He has a budget of $1,000 and his plot has a maximum of 100 square feet. The apple trees he wants cost $40 each and require 5 square feet. The cherry trees he wants cost $50 each and require 4 square feet.

The following first-quadrant graph shows the system of inequalities that represents the situation,
{5x+4y≤100
40x+50y≤1000,
with the number of apple trees along the x-axis and the number of cherry trees along the y-axis.


Graph labeled as in text; shaded below 1 solid segment from (0, 20) to (25, 0); shaded below second solid segment from (0, 25) to (20, 0); & solution below both segments.

© 2017 StrongMind. Created using GeoGebra.


Which combinations of trees are solutions for this situation?

Select all answers that apply.

Responses

​0 apple trees and 25 cherry trees
14 apple trees and 11 cherry trees
12 apple trees and 6 cherry trees
​24 apple trees and 0 cherry trees
4 apple trees and 16 cherry trees
20 apple trees and 4 cherry trees

Answer 1

The valid combinations of trees that are solutions to the given system of inequalities are: 14 apple trees and 11 cherry trees, 12 apple trees and 6 cherry trees, 24 apple trees and 0 cherry trees, 4 apple trees and 16 cherry trees, and 20 apple trees and 4 cherry trees.

Step-by-step explanation:

The graph represents a system of inequalities that Anton must satisfy in order to plant the desired number of apple and cherry trees within his budget and plot size constraints. The shaded region below the first solid segment and below the second solid segment represents the solutions to the system of inequalities. To determine which combinations of trees are solutions, we need to determine which points fall within the shaded region. Let's evaluate each option:

1. 0 apple trees and 25 cherry trees: This combination satisfies the inequalities since 5(0) + 4(25) = 100 (less than or equal to 100) and 40(0) + 50(25) = 1250 (greater than 1000), so this is not a valid solution.
2. 14 apple trees and 11 cherry trees: This combination satisfies the inequalities since 5(14) + 4(11) = 94 (less than or equal to 100) and 40(14) + 50(11) = 890 (less than or equal to 1000), so this is a valid solution.
3. 12 apple trees and 6 cherry trees: This combination satisfies the inequalities since 5(12) + 4(6) = 78 (less than or equal to 100) and 40(12) + 50(6) = 600 (less than or equal to 1000), so this is a valid solution.
4. 24 apple trees and 0 cherry trees: This combination satisfies the inequalities since 5(24) + 4(0) = 120 (less than or equal to 100) and 40(24) + 50(0) = 960 (less than or equal to 1000), so this is a valid solution.
5. 4 apple trees and 16 cherry trees: This combination satisfies the inequalities since 5(4) + 4(16) = 84 (less than or equal to 100) and 40(4) + 50(16) = 880 (less than or equal to 1000), so this is a valid solution.
6. 20 apple trees and 4 cherry trees: This combination satisfies the inequalities since 5(20) + 4(4) = 116 (less than or equal to 100) and 40(20) + 50(4) = 1000 (equal to 1000), so this is a valid solution.

The valid combinations/solutions are: