A manufacturer produces a standard model and a deluxe model of a 25-inch (in). television set. the standard model requires 12 h of labor to produce, and the deluxe model requires 18 h. The company has 360 h of labor available per week. the plant's capacity is a total of 25 sets per week. if all the available time and capacity are to be used, how many of each type of set should be produced?
s = standards
d = deluxe
12 s +18 d = 360 ____divide by 6
s + d = 25 ________multiply by 2
2 s + 3 d = 60
2 s + 2 d = 50
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d = 10
then s = 15
A manufacturer company produces a standard model and deluxe model of 25”
TV. The standard model requires 12th of Labor to produce and the deluxe modal
requires 18th. The company has 360th of labor available per week. The plant's
capacity is a total of 25 sets per week. If all the available time and capacity are to be
used, how many of each type of set should be produced.
To determine the optimal number of each type of set to produce, we can use a system of linear equations.
Let's assume that the number of standard sets produced is x, and the number of deluxe sets produced is y.
Based on the given information, we can establish the following equations:
1. Labor Equation: 12x + 18y = 360
(The total labor used by producing standard sets is 12x hours, and the total labor used by producing deluxe sets is 18y hours. The total labor available is 360 hours.)
2. Capacity Equation: x + y = 25
(The total number of standard and deluxe sets should not exceed the plant's capacity, which is 25 sets.)
We now have a system of two linear equations. We can solve this system to find the values of x and y.
Step 1: Rearrange the Capacity Equation to solve for x:
x = 25 - y
Step 2: Substitute the value of x from Step 1 into the Labor Equation:
12(25 - y) + 18y = 360
300 - 12y + 18y = 360
6y = 60
y = 10
Step 3: Substitute the value of y back into the Capacity Equation:
x + 10 = 25
x = 15
Therefore, the optimal solution is to produce 15 standard sets and 10 deluxe sets.
To find out how many of each type of set should be produced, we need to set up a system of equations based on the given information.
Let's assume the number of standard models produced is "x" and the number of deluxe models produced is "y".
The total amount of labor required for the standard models is 12 hours per set, so the total labor used for producing "x" standard models is 12x hours.
Similarly, the total amount of labor required for the deluxe models is 18 hours per set, so the total labor used for producing "y" deluxe models is 18y hours.
According to the given information, the total labor available per week is 360 hours. So, we have the equation:
12x + 18y = 360 (Equation 1)
The plant's capacity is a total of 25 sets per week. So, we have the equation:
x + y = 25 (Equation 2)
Now, we have a system of two equations with two variables. We can solve these equations simultaneously to find the values of "x" and "y".
Let's solve Equation 2 for "x" and substitute it into Equation 1:
x = 25 - y
12(25 - y) + 18y = 360
300 - 12y + 18y = 360
6y = 60
y = 10
Substituting the value of "y" back into Equation 2:
x + 10 = 25
x = 15
Therefore, to utilize all available labor and capacity, the manufacturer should produce 15 standard models and 10 deluxe models.