The table below shows the number of hours required in each of two departments to make one unit of various products A, B and C. For example, product B requires 1 hour of time in department I and 3 hours in department II.
HOURS REQUIRED PER UNIT OF PRODUCT
DEPARTMENT 1
A=1
B=1
C=9
DEPARTMENT 2
A=1
B=3
C=7
Find the number of units of A, B, and C which could be made if department I has 75 hours available and Department II has 65 hours available. It is necessary that all of the available hours be used.
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solution.
To find the number of units of A, B, and C that can be made, we need to determine the maximum number of units that can be produced given the available hours in each department.
First, let's calculate the number of units that can be produced in Department 1 (D1) and Department 2 (D2) separately.
For Department 1:
- A requires 1 hour per unit, and we have 75 hours available. Therefore, we can produce a maximum of 75 units of A in Department 1.
- B requires 1 hour per unit, and we have 75 hours available. Therefore, we can produce a maximum of 75 units of B in Department 1.
- C requires 9 hours per unit, and we have 75 hours available. Therefore, we can produce a maximum of 75/9 = 8.33 units of C in Department 1. Since we can't produce fractions of a unit, we can produce a maximum of 8 units of C in Department 1.
For Department 2:
- A requires 1 hour per unit, and we have 65 hours available. Therefore, we can produce a maximum of 65 units of A in Department 2.
- B requires 3 hours per unit, and we have 65 hours available. Therefore, we can produce a maximum of 65/3 = 21.67 units of B in Department 2. Since we can't produce fractions of a unit, we can produce a maximum of 21 units of B in Department 2.
- C requires 7 hours per unit, and we have 65 hours available. Therefore, we can produce a maximum of 65/7 = 9.29 units of C in Department 2. Since we can't produce fractions of a unit, we can produce a maximum of 9 units of C in Department 2.
Now let's compare the maximum number of units that can be produced in each department for each product:
- Department 1:
- A: 75 units
- B: 75 units
- C: 8 units
- Department 2:
- A: 65 units
- B: 21 units
- C: 9 units
To determine the total number of units that can be made, we need to consider the maximum number of units that can be produced in both departments for each product. Since we must use all the available hours, we will take the minimum value from the two departments for each product.
Therefore, the maximum number of units of each product that can be made is as follows:
- A: 65 units (minimum value between Department 1 and Department 2)
- B: 21 units (minimum value between Department 1 and Department 2)
- C: 8 units (minimum value between Department 1 and Department 2)
So, with the given available hours in each department, the maximum number of units that can be produced is 65 units of A, 21 units of B, and 8 units of C.