A farmer plans to purchase some chickens and some goats. The farmer decides to spend at least $100 and will purchase at least 8 animals. Each chicken costs $5 and each goat costs $25.
If we let x = number of chickens purchased and y = number of goats purchased, graph the solution set of the system of inequalities. Determine the coordinates of the corner points.
I dont understand how to graph this
A farmer plans to purchase some chickens and some goats. The farmer decides to spend at least $100 and will purchase at least 8 animals. Each chicken costs $5 and each goat costs $25.
If we let x = number of chickens purchased and y = number of goats purchased, graph the solution set of the system of inequalities. Determine the coordinates of the corner points.
I dont understand how to graph this
Each chicken costs $5 and each goat costs $25.
So the total cost is 5x+ 25y.
The farmer decides to spend at least $100
So $5x+ 25y ≥ 100$. The region satisfying that is bounded by the equation $5x+ 2y= 100$. When x= 0, 25y= 100 so y= 4. One point on the line is (0, 4). When y= 0, 5x= 100 so x= 20. Another point on the line is (20, 0). The graph of the line is the straight line from (0, 4) to (20, 0). x and y are numbers of animals so cannot be negative. The region satisfying $5x+ 25y ≥ 100$ is the region in the first quadrant above that line.
and will purchase at least 8 animals
So \(\displaystyle x+ y ≥ 8\). The region satisfying that is bounded by the equation $x+ y= 8$. When x= 0, y= 8 so (0, 8) is a point on that line. When y= 0, x= 8 so (0, 8) is a point on that line. The line $x+ y= 8$ is the line from (0, 8) to (8, 0). The region satisfying $x+ y ≥ 8$ is the region above that line. The region satisfying all of these is the region in the first quadrant above both of those lines.
It should be clear that (0, 8) and (20, 0) are vertices of that region. The third vertex is the point of intersection of the two lines so where both $5x+ 25y= 100$, which is the same as $x+ 5y= 20$, and $x+ y= 8$. From the second, $y= 8- x$ so $x+ 5y= x+ 5(8- x)= -4x+ 40= 20$ so $-4x= -20$, $x= 5$. $y= 8- x= 8- 5= 3$. The third vertex is (5, 3).
The farmer decides to spend at least $100 and will purchase at least 8 animals
So $5x+ 25y ≥ 100$. The region satisfying that is bounded by the equation $5x+ 2y= 100$. When x= 0, 25y= 100 so y= 4. One point on the line is (0, 4). When y= 0, 5x= 100 so x= 20. Another point on the line is (20, 0). The graph of the line is the straight line from (0, 4) to (20, 0). x and y are numbers of animals so cannot be negative. The region satisfying $5x+ 25\ge 100$ is the region in the first quadrant above that line.
