Determine the equations which ensure the weight capacity in an airlift and budget in hiring programmer analysts.
Understand the Problem
The question involves determining equations related to logistical constraints for airlifting food and hiring personnel for a reservation system. Specifically, it asks for equations based on weight and volume capacities and budgetary limits.
Answer
The equations for constraints are: Weight: \( 120x_1 + 300x_2 + 250x_3 + 500x_4 \leq 60000 \); Budget: \( c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 \leq \text{Budget} \).
Answer for screen readers
The equations that ensure the weight capacity and budget constraints are met are:
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Weight constraint: $$ 120x_1 + 300x_2 + 250x_3 + 500x_4 \leq 60000 $$
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Budget constraint: $$ c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 \leq \text{Budget} $$
Steps to Solve
- Identify Variables for Weight Capacity
Let ( x_1, x_2, x_3, x_4 ) be the quantities of the four types of food. The weights of the food items are given as 120, 300, 250, and 500 pounds respectively. The equation for weight capacity can thus be expressed as:
$$ 120x_1 + 300x_2 + 250x_3 + 500x_4 \leq 60000 $$
This equation ensures that the total weight does not exceed the weight capacity of 60,000 pounds.
- Identify Variables for Budget Constraints
Assuming each type of food has a cost per unit, let ( c_1, c_2, c_3, c_4 ) represent the costs associated with ( x_1, x_2, x_3, x_4 ) respectively. We can formulate the budget constraint as:
$$ c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 \leq \text{Budget} $$
This equation ensures that the total cost of the food does not exceed the available budget.
- Combine Equations for Overall Constraints
To ensure both weight capacity and budget constraints are satisfied simultaneously, we can represent them together as a set of inequalities:
$$ \begin{align*} 120x_1 + 300x_2 + 250x_3 + 500x_4 & \leq 60000 \ c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 & \leq \text{Budget} \end{align*} $$
The equations that ensure the weight capacity and budget constraints are met are:
-
Weight constraint: $$ 120x_1 + 300x_2 + 250x_3 + 500x_4 \leq 60000 $$
-
Budget constraint: $$ c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 \leq \text{Budget} $$
More Information
These equations are essential for resource allocation in logistical operations, such as airlifting food supplies. By constructing these inequalities, planners can optimize their shipments to fit within physical and financial limits.
Tips
- Failing to account for all variables when forming the equations.
- Confusing weight and cost units; ensure consistency (pounds for weight, dollars for cost).
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