Question

Liam wants to plant a small orchard with apples and cherries in a 62-square-foot plot, and he has a budget of $345 for purchasing trees. The apple trees he wants cost $30 each and require 8 square feet each for maximum fruit production. The cherry trees he wants cost $45 each and require 6 square feet for maximum fruit production. If he sells all the fruit, he is able to earn $90 per apple tree and $120 per cherry tree. The system of inequalities that represents this situation, (30x+45y≤345) (8x+6y≤62), is shown graphed in the first quadrant with the number of apple trees along the x-axis and the number of cherry trees along the y-axis. Which points are the extreme points for the solution set for this situation?

Select all that apply.
a) (7.75,0)
b) (11.5,0)
c) (0,7.67)
d) (0,0)
e) (0,10.33)
f) (4,5)

Answer 1

The extreme points for the solution set to the system of inequalities are (7.75,0), (11.5,0), (0,7.67), (0,0), (0,10.33), and (4,5).the correct options are (a), (b), (c), (d), (e), and (f).

Step-by-step explanation:

The system of inequalities that represents this situation is:

30x + 45y ≤ 345 (equation 1)

8x + 6y ≤ 62 (equation 2)

To find the extreme points of the solution set, we need to solve the system of inequalities graphically. We can plot the two equations on a graph and find the points where the lines intersect or touch the axes.

The extreme points for the solution set are:

(7.75,0)

(11.5,0)

(0,7.67)

(0,0)

(0,10.33)

(4,5)

Therefore, the correct options are (a), (b), (c), (d), (e), and (f).