Question

A local company makes two types of soups, chicken and beef barley. Each batch of chicken soup takes 3 hours to prepare and 4 hours to package. Each batch of beef barley takes 3.5 hours to prepare but only 2 hours to package. There are 5 preparation workers and 6 packaging workers in the company. Each of them works 40 hours per week.

Part A:
Create a system of two inequalities that relates the number of batches of chicken soup, c, and the number of batches of beef barley soup, b, that can be made by the 5 preparation workers and the 6 packaging workers each week. Assume c≥0 and b≥0.

Part B:
Give two possible combinations of batches of chicken soup and beef barley soup that could be made in one week based on the constraints?

Answer 1

Explanation:

Part A:

Let's use the information given in the problem to create the inequalities:

Each batch of chicken soup takes 3 hours to prepare, so the total preparation time for c batches of chicken soup is 3c hours.

Each batch of beef barley takes 3.5 hours to prepare, so the total preparation time for b batches of beef barley soup is 3.5b hours.

Each batch of chicken soup takes 4 hours to package, so the total packaging time for c batches of chicken soup is 4c hours.

Each batch of beef barley takes 2 hours to package, so the total packaging time for b batches of beef barley soup is 2b hours.

There are 5 preparation workers who work 40 hours per week, so the total preparation time available is 5 workers x 40 hours/worker = 200 hours.

There are 6 packaging workers who work 40 hours per week, so the total packaging time available is 6 workers x 40 hours/worker = 240 hours.

To create the system of inequalities, we need to make sure that the total preparation and packaging times for both types of soup do not exceed the available time. Therefore:

3c + 3.5b ≤ 200 (total preparation time ≤ available preparation time)

4c + 2b ≤ 240 (total packaging time ≤ available packaging time)

Also, we have the non-negativity constraints:

c ≥ 0 (number of batches of chicken soup is non-negative)

b ≥ 0 (number of batches of beef barley soup is non-negative)

Therefore, the system of inequalities is:

3c + 3.5b ≤ 200

4c + 2b ≤ 240

c ≥ 0

b ≥ 0

Part B:

To find two possible combinations of batches of chicken soup and beef barley soup that could be made in one week based on the constraints, we can try different values for c and b that satisfy the inequalities from Part A. Here are two possible combinations:

Combination 1: c = 40, b = 0

Total preparation time: 3c + 3.5b = 3(40) + 3.5(0) = 120

Total packaging time: 4c + 2b = 4(40) + 2(0) = 160

Both inequalities are satisfied, and c and b are non-negative.

Therefore, in one week, the company can make 40 batches of chicken soup and 0 batches of beef barley soup.

Combination 2: c = 30, b = 20

Total preparation time: 3c + 3.5b = 3(30) + 3.5(20) = 135

Total packaging time: 4c + 2b = 4(30) + 2(20) = 160

Both inequalities are satisfied, and c and b are non-negative.

Therefore, in one week, the company can make 30 batches of chicken soup and 20 batches of beef barley soup.