To represent the constraints given in the problem, we can establish a series of inequalities based on the information provided. Let's break down the requirements:
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Cost Constraint: The total cost of the volleyballs and soccer balls cannot exceed $1,100.
- The cost for volleyballs = \(20x\) (where \(x\) is the number of volleyballs).
- The cost for soccer balls = \(25y\) (where \(y\) is the number of soccer balls).
- The inequality representing this constraint is: \[ 20x + 25y \leq 1100 \]
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Quantity Constraint: The coach plans to order at least 50 volleyballs and soccer balls in total.
- This can be written as: \[ x + y \geq 50 \]
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Non-negativity Constraint: The number of volleyballs and soccer balls cannot be negative: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]
Graphing the Constraints
To graph these inequalities, we can rewrite them in slope-intercept form:
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From the cost constraint: \[ 20x + 25y \leq 1100 \quad \text{or} \quad 25y \leq 1100 - 20x \quad \text{or} \quad y \leq \frac{1100 - 20x}{25} \] Simplifying: \[ y \leq 44 - 0.8x \]
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From the quantity constraint: \[ x + y \geq 50 \quad \text{or} \quad y \geq 50 - x \]
Graph Features
- The line \(y = 44 - 0.8x\) represents the upper boundary of the spending constraint. It slopes downward.
- The line \(y = 50 - x\) represents the lower boundary of the quantity constraint. It slopes downward as well.
Finding the Feasible Region
To determine the feasible region (where the solutions to the constraints are satisfied), you need to:
- Graph the two lines: \(y = 44 - 0.8x\) and \(y = 50 - x\).
- Shade below the line \(y = 44 - 0.8x\) (since it is less than or equal to).
- Shade above the line \(y = 50 - x\) (since it is greater than or equal to).
- The feasible region will be the area where the shadings overlap, while also considering that \(x \geq 0\) and \(y \geq 0\).
Vertices of the Feasible Region
The vertices of the feasible region can typically be calculated by solving the system of equations formed by the equality of the two lines:
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Set \(44 - 0.8x = 50 - x\): \[ 0.2x = 6 \quad \Rightarrow \quad x = 30 \] Substitute \(x = 30\) into either equation to find \(y\): \[ y = 50 - 30 = 20 \]
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Find the intersection with the axes and any additional points of intersection to fully define the vertex set (check intercepts).
Conclusion
The feasible regions will be bounded by these lines and should include all combinations of volleyballs and soccer balls that meet the coach's requirements. The vertices found can be used to determine which combinations maximize or minimize the number of balls purchased while adhering to the constraints.