A manufacturer makes two products A and B, each of which is processed in two departments, production and finishing. Each A takes 3 hours to produce and six hours to finish, whereas each B takes 5 hours to produce and 2 hours to finish. How many units of A and B can be produced and finished if exactly 24 hours are available in each department and all hours must be used?
To find the maximum number of units of products A and B that can be produced and finished given the time constraints, let's break down the problem into steps:
Step 1: Calculate the available hours in each department.
- In the production department, there are 24 hours available.
- In the finishing department, there are also 24 hours available.
Step 2: Determine the number of units of product A that can be produced and finished.
- Each unit of A takes 3 hours to produce and 6 hours to finish, which totals to 9 hours.
- Divide the available hours in the production department (24 hours) by the total time required for each unit of A (9 hours).
- 24 hours / 9 hours = 2.67 (approximately)
Since we cannot produce and finish fractions of units, we need to round down to the nearest whole number.
- The maximum number of units of A that can be produced and finished is 2.
Step 3: Determine the number of units of product B that can be produced and finished.
- Each unit of B takes 5 hours to produce and 2 hours to finish, which totals to 7 hours.
- Divide the available hours in the production department (24 hours) by the total time required for each unit of B (7 hours).
- 24 hours / 7 hours = 3.43 (approximately)
Again, we must round down to the nearest whole number.
- The maximum number of units of B that can be produced and finished is 3.
Therefore, in the given time constraints, we can produce and finish 2 units of product A and 3 units of product B.
Let's start by defining the variables:
Let's assume the number of units of product A produced as "x" and number of units of product B produced as "y".
Now, let's calculate the time required for producing and finishing the units:
Product A:
- The time required to produce "x" units of product A is 3 * x hours.
- The time required to finish "x" units of product A is 6 * x hours.
Product B:
- The time required to produce "y" units of product B is 5 * y hours.
- The time required to finish "y" units of product B is 2 * y hours.
We know that exactly 24 hours are available in each department, so we can set up the following equations:
Production Department:
3x + 5y = 24
Finishing Department:
6x + 2y = 24
We can solve these equations using any algebraic method, such as substitution or elimination. Let's use substitution method here:
From the first equation, we can isolate x:
3x = 24 - 5y
x = (24 - 5y) / 3
Now, substitute this value of x in the second equation:
6 * ((24 - 5y) / 3) + 2y = 24
Simplifying the equation:
12y - 10y + 2y = 72
4y = 72
y = 72 / 4
y = 18
Now substitute the value of y in the first equation to find x:
3x + 5 * 18 = 24
3x + 90 = 24
3x = 24 - 90
3x = -66
x = -66 / 3
x = -22
Since you cannot have negative units, it seems there is no solution that meets all the conditions of the problem.