Question

problem: after a hurricane, a humanitarian group is sending food and clothing to victims. each food container is estimated to feed 12 people and each clothing container is estimated to help 5 people. organizers want to determine the number of containers of food and clothing to maximize the number of victims helped.
step 1 – writing the objective function
in this problem, you want to determine the maximum/minimum (circle one) of
_____________________________
if x represents the number of containers of food and y represents the number of containers of clothing, write an equation that expresses the number n of victims to be helped in terms of x and y. this is the objective function.
objective function: _____________________________
step 2 – determining the constraints
the aircraft that ship the containers have weight and volume restrictions. such limitations are called constraints. each constraint must be translated into an inequality in order to form a system of linear inequalities.
the weight and volume restrictions on the aircraft are as follows:

• total weight cannot exceed 19,000 pounds.
• total volume must be no more than 8,000 cubic feet.

each food container weighs 50 pounds and occupies 20 cubic feet. each clothing container weighs 20 pounds and occupies 10 cubic feet.
a. write the aircraft’s weight constraint as a linear inequality (remember: x represents # of container of food and y represents # of containers of clothing).
b. write the aircraft’s volume constraint as a linear inequality.
c. the inequalities x≥0 and y≥0 are often additional constraints. explain what these two inequalities mean in the context of the problem.

Explanation:

Step 1: Determine Maximization/Minimization

The goal is to maximize the number of victims helped (since we want to help as many as possible). So we circle "maximum". Then, for the objective function: each food container (x) helps 12 people, each clothing container (y) helps 5 people. So the number of victims \( N = 12x + 5y \).

Step 2a: Weight Constraint

Each food container (x) weighs 50 pounds, each clothing container (y) weighs 20 pounds. Total weight ≤ 19,000 pounds. So the inequality is \( 50x + 20y \leq 19000 \).

Step 2b: Volume Constraint

Each food container (x) occupies 20 cubic feet, each clothing container (y) occupies 10 cubic feet. Total volume ≤ 8,000 cubic feet. So the inequality is \( 20x + 10y \leq 8000 \).

Step 2c: Explain \( x \geq 0 \) and \( y \geq 0 \)

\( x \geq 0 \) means the number of food containers can't be negative (we can't have a negative number of containers). Similarly, \( y \geq 0 \) means the number of clothing containers can't be negative.

Answer:

Step 1:

• Circle: maximum
• Objective Function: \( N = 12x + 5y \)

Step 2a:

Weight Constraint: \( \boldsymbol{50x + 20y \leq 19000} \)

Step 2b:

Volume Constraint: \( \boldsymbol{20x + 10y \leq 8000} \)

Step 2c:

\( x \geq 0 \) means the number of food containers cannot be negative (we can’t have a negative number of food containers). \( y \geq 0 \) means the number of clothing containers cannot be negative (we can’t have a negative number of clothing containers).