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Sagot :
To determine the feasible region representing the possible combinations of coffee and tea for the order, we need to graph the system of inequalities derived from the problem statement.
Here are the basic steps we follow:
1. Identify the Inequalities:
We have two inequalities from the given problem:
[tex]\[ 6c + 13t \leq 3500 \quad \text{(Cost Inequality)} \][/tex]
[tex]\[ 3.50c + 4t \geq 1900 \quad \text{(Profit Inequality)} \][/tex]
2. Determine the Boundary Lines:
We solve for the boundary lines by setting each inequality to equality.
For [tex]\(6c + 13t = 3500\)[/tex]:
- When [tex]\(c = 0\)[/tex]:
[tex]\[ 13t = 3500 \implies t = \frac{3500}{13} \approx 269.2307692307692 \][/tex]
- When [tex]\(t = 0\)[/tex]:
[tex]\[ 6c = 3500 \implies c = \frac{3500}{6} \approx 583.3333333333334 \][/tex]
For [tex]\(3.50c + 4t = 1900\)[/tex]:
- When [tex]\(c = 0\)[/tex]:
[tex]\[ 4t = 1900 \implies t = \frac{1900}{4} = 475 \][/tex]
- When [tex]\(t = 0\)[/tex]:
[tex]\[ 3.50c = 1900 \implies c = \frac{1900}{3.50} \approx 542.8571428571429 \][/tex]
3. Plot the Boundary Lines:
On a coordinate system with [tex]\(c\)[/tex] (pounds of coffee) on the horizontal axis and [tex]\(t\)[/tex] (pounds of tea) on the vertical axis:
- Plot the line [tex]\(6c + 13t = 3500\)[/tex] using the points [tex]\((0, 269.2307692307692)\)[/tex] and [tex]\((583.3333333333334, 0)\)[/tex].
- Plot the line [tex]\(3.50c + 4t = 1900\)[/tex] using the points [tex]\((0, 475)\)[/tex] and [tex]\((542.8571428571429, 0)\)[/tex].
4. Determine the Feasible Region:
- For the inequality [tex]\(6c + 13t \leq 3500\)[/tex], the feasible region is below the line [tex]\(6c + 13t = 3500\)[/tex].
- For the inequality [tex]\(3.50c + 4t \geq 1900\)[/tex], the feasible region is above the line [tex]\(3.50c + 4t = 1900\)[/tex].
5. Shade the Intersection of the Feasible Regions:
The final feasible region on the graph is the area where the regions defined by the two inequalities overlap. This represents the possible combinations of [tex]\(c\)[/tex] and [tex]\(t\)[/tex] that maximize the shop's constraints on cost and profit.
In conclusion, the shaded region representing the possible combinations of coffee and tea for this order is the intersection of the regions below the cost line [tex]\(6c + 13t = 3500\)[/tex] and above the profit line [tex]\(3.50c + 4t = 1900\)[/tex].
Here are the basic steps we follow:
1. Identify the Inequalities:
We have two inequalities from the given problem:
[tex]\[ 6c + 13t \leq 3500 \quad \text{(Cost Inequality)} \][/tex]
[tex]\[ 3.50c + 4t \geq 1900 \quad \text{(Profit Inequality)} \][/tex]
2. Determine the Boundary Lines:
We solve for the boundary lines by setting each inequality to equality.
For [tex]\(6c + 13t = 3500\)[/tex]:
- When [tex]\(c = 0\)[/tex]:
[tex]\[ 13t = 3500 \implies t = \frac{3500}{13} \approx 269.2307692307692 \][/tex]
- When [tex]\(t = 0\)[/tex]:
[tex]\[ 6c = 3500 \implies c = \frac{3500}{6} \approx 583.3333333333334 \][/tex]
For [tex]\(3.50c + 4t = 1900\)[/tex]:
- When [tex]\(c = 0\)[/tex]:
[tex]\[ 4t = 1900 \implies t = \frac{1900}{4} = 475 \][/tex]
- When [tex]\(t = 0\)[/tex]:
[tex]\[ 3.50c = 1900 \implies c = \frac{1900}{3.50} \approx 542.8571428571429 \][/tex]
3. Plot the Boundary Lines:
On a coordinate system with [tex]\(c\)[/tex] (pounds of coffee) on the horizontal axis and [tex]\(t\)[/tex] (pounds of tea) on the vertical axis:
- Plot the line [tex]\(6c + 13t = 3500\)[/tex] using the points [tex]\((0, 269.2307692307692)\)[/tex] and [tex]\((583.3333333333334, 0)\)[/tex].
- Plot the line [tex]\(3.50c + 4t = 1900\)[/tex] using the points [tex]\((0, 475)\)[/tex] and [tex]\((542.8571428571429, 0)\)[/tex].
4. Determine the Feasible Region:
- For the inequality [tex]\(6c + 13t \leq 3500\)[/tex], the feasible region is below the line [tex]\(6c + 13t = 3500\)[/tex].
- For the inequality [tex]\(3.50c + 4t \geq 1900\)[/tex], the feasible region is above the line [tex]\(3.50c + 4t = 1900\)[/tex].
5. Shade the Intersection of the Feasible Regions:
The final feasible region on the graph is the area where the regions defined by the two inequalities overlap. This represents the possible combinations of [tex]\(c\)[/tex] and [tex]\(t\)[/tex] that maximize the shop's constraints on cost and profit.
In conclusion, the shaded region representing the possible combinations of coffee and tea for this order is the intersection of the regions below the cost line [tex]\(6c + 13t = 3500\)[/tex] and above the profit line [tex]\(3.50c + 4t = 1900\)[/tex].
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