A company produces doll houses and sets of doll furniture. The doll houses take 3 hours of labor to produce, and the furniture sets take 8 hours. The labor available is limited to 400 hours per week, and the total production capacity is 100 items per week. Existing orders require that at least 20 doll houses and 10 sets of furniture be produced per week. Write a system of inequalities representing this situation, where x is the number of doll houses and y is the number of furniture sets.
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To represent this situation as a system of inequalities, we need to consider the constraints given:
1. Labor constraint: The labor required to produce doll houses and furniture sets must not exceed 400 hours.
2. Production capacity constraint: The total number of items produced (doll houses + furniture sets) must not exceed 100 per week.
3. Order requirement constraint: At least 20 doll houses and 10 furniture sets need to be produced per week.
Let's define the variables:
x = number of doll houses produced
y = number of furniture sets produced
Now, let's write the inequalities:
1. Labor Constraint:
The total labor required to produce x doll houses and y furniture sets should not exceed 400 hours.
Labor required for doll houses: 3 hours per doll house
Labor required for furniture sets: 8 hours per furniture set
So, the labor constraint inequality is:
3x + 8y ≤ 400
2. Production Capacity Constraint:
The total number of items produced (x doll houses + y furniture sets) should not exceed 100.
So, the production capacity constraint inequality is:
x + y ≤ 100
3. Order Requirement Constraint:
To satisfy the existing orders, we need to produce at least 20 doll houses and 10 furniture sets.
So, the order requirement constraint inequalities are:
x ≥ 20
y ≥ 10
Therefore, the system of inequalities representing this situation is:
3x + 8y ≤ 400
x + y ≤ 100
x ≥ 20
y ≥ 10