A company produces three types of color TVs: Models X, Y and Z. Each model X requires 2 hours of electronic work and 2 hours of assembly work. Each Y model requires 1 hour of electronic work and 3 hours of assembly work. Each model Z requires 3 hours of electronic work and 2 hours of assembly work. They have available a 100 hours of work per week, for each type of work, electronic and assembly. How many televisions of each model should be produced each week if they use all available time?
To determine the number of televisions of each model that should be produced each week, we need to set up a system of equations based on the time required for electronic and assembly work.
Let's assume that the number of X model televisions produced each week is denoted by 'x', Y model televisions by 'y', and Z model televisions by 'z'.
Based on the given information, we can set up the following equations:
Equation 1: 2x + y + 3z = 100 (represents the total electronic work available)
Equation 2: 2x + 3y + 2z = 100 (represents the total assembly work available)
Now, let's solve this system of equations using either substitution or elimination method:
Using the substitution method:
1. Solve Equation 1 for y: y = 100 - 2x - 3z
2. Substitute the value of y in Equation 2: 2x + 3(100 - 2x - 3z) + 2z = 100
3. Simplify the equation: 2x + 300 - 6x - 9z + 2z = 100
4. Combine like terms: -4x - 7z = -200
Now you can solve this equation for one of the variables (either x or z) in terms of the other variable.
Let's solve for z by expressing it in terms of x:
-4x - 7z = -200
-7z = -200 + 4x
z = (200 - 4x) / 7
Now you can substitute this value of z back into either Equation 1 or Equation 2 to solve for x.
Let's substitute it back into Equation 1:
2x + y + 3((200 - 4x) / 7) = 100
Multiply both sides by 7 to eliminate the fraction:
14x + 7y + 600 - 12x = 700
Combine like terms: 2x + 7y = 100
Now we have another equation in terms of x and y. We can solve this equation simultaneously with Equation 1 or Equation 2 to find the values of x and y.
At this point, we have two equations:
2x + 7y = 100 (Equation A)
2x + y + 3z = 100 (Equation 1)
Using substitution or elimination method, you can solve these equations to find the values of x, y, and z.
To determine the number of televisions of each model that should be produced each week, we need to use the available time efficiently and satisfy the work hour requirements for each model.
Let's solve this problem step by step:
Step 1: Define the variables.
Let's assume the number of X, Y, and Z models to be produced each week as x, y, and z respectively.
Step 2: Formulate the constraints.
We know that each X model requires 2 hours of electronic work and 2 hours of assembly work.
So, the constraint for electronic work can be written as:
2x + 1y + 3z ≤ 100
And the constraint for assembly work can be written as:
2x + 3y + 2z ≤ 100
Step 3: Define the objective function.
Since we want to maximize the number of televisions produced, we can set the objective function as:
Maximize: x + y + z
Step 4: Solve the linear programming problem.
Using the constraints and the objective function, we can solve this linear programming problem using various methods like the graphical method, simplex method, or software tools like Excel Solver or MATLAB's linprog function.
Solving the problem will yield the optimal values of x, y, and z, which represent the number of X, Y, and Z models to be produced each week to use all available time efficiently.
Note: The exact solution will depend on the concrete values of the coefficients and the constraints.