The Red Cross wants to airlift supplies into a South American country which has experienced an earthquake. Four types of supplies are being considered. One container of a particula... The Red Cross wants to airlift supplies into a South American country which has experienced an earthquake. Four types of supplies are being considered. One container of a particular item weighs 120, 300, 250, and 500 pounds, respectively, for the four items. If the airplane to be used has a weight capacity of 60,000 pounds and x equals the number of containers shipped of item j: (a) Determine the equation which ensures that the plane will be loaded to its weight capacity. (b) If it is decided to devote this plane to one supply item only, how many containers could be shipped of each item? Read more

Steps to Solve

1. Identify Variables and Constants

Define the variables for the number of containers for each supply type:

• Let $x_1$ be the number of containers for item 1 (120 lbs)
• Let $x_2$ be the number of containers for item 2 (300 lbs)
• Let $x_3$ be the number of containers for item 3 (250 lbs)
• Let $x_4$ be the number of containers for item 4 (500 lbs)

The airplane has a weight capacity of 60,000 pounds.

2. Formulate the Weight Capacity Equation

The total weight of the containers should not exceed the weight capacity of the airplane. The equation can be set up as follows:

$$ 120x_1 + 300x_2 + 250x_3 + 500x_4 \leq 60000 $$

3. Calculate Maximum Containers for Each Item

Now, if the plane is to carry only one type of supply item, we can calculate the maximum number of containers that can be shipped for each item.

• For item 1 (120 lbs):

  $$ x_1 = \frac{60000}{120} = 500 $$

• For item 2 (300 lbs):

  $$ x_2 = \frac{60000}{300} = 200 $$

• For item 3 (250 lbs):

  $$ x_3 = \frac{60000}{250} = 240 $$

• For item 4 (500 lbs):

  $$ x_4 = \frac{60000}{500} = 120 $$