a paint factory produces oil paint and latex paint. at least 2 l of oil paint is produced for every 1 l of…

a paint factory produces oil paint and latex paint. at least 2 l of oil paint is produced for every 1 l of latex paint. in a day the factory can produce up to 6 thousand litres of oil paint and up to 8 thousand litres of latex paint. the factory sells latex paint for $8 per litre and oil paint for $10 per litre. determine the daily combination of latex and oil paint that will maximize the revenue. x = number of l of latex paint

a paint factory produces oil paint and latex paint. at least 2 l of oil paint is produced for every 1 l of latex paint. in a day the factory can produce up to 6 thousand litres of oil paint and up to 8 thousand litres of latex paint. the factory sells latex paint for $8 per litre and oil paint for $10 per litre. determine the daily combination of latex and oil paint that will maximize the revenue. x = number of l of latex paint

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of litres of latex - paint and $y$ be the number of litres of oil - paint.

Step2: Set up constraints

  1. Production capacity constraints:
    • $y\leq6000$ (maximum production of oil - paint is 6000 litres)
    • $x\leq8000$ (maximum production of latex - paint is 8000 litres)
    • $y\geq2x$ (at least 2L of oil - paint is produced for every 1L of latex - paint)
    • $x\geq0,y\geq0$ (non - negative production)

Step3: Set up the objective function

The revenue function $R$ is given by $R = 8x+10y$ (latex paint sells for $8 per litre and oil paint sells for $10 per litre).

Step4: Find the feasible region

The intersection of the inequalities $y\leq6000$, $x\leq8000$, $y\geq2x$, $x\geq0,y\geq0$ gives the feasible region. The corner - points of the feasible region are found by solving the systems of equations:

  • Intersection of $y = 2x$ and $y=6000$: Substitute $y = 6000$ into $y = 2x$, we get $x = 3000$. So the point is $(3000,6000)$.
  • Intersection of $y = 2x$ and $x = 0$ is the origin $(0,0)$.
  • Intersection of $x = 8000$ and $y = 2x$ gives $y = 16000$, but since $y\leq6000$, this part of the line $y = 2x$ is not in the feasible region.
  • Intersection of $x = 8000$ and $y = 6000$ gives the point $(8000,6000)$.

Step5: Evaluate the objective function at corner - points

  • At $(0,0)$: $R=8\times0 + 10\times0=0$.
  • At $(3000,6000)$: $R=8\times3000+10\times6000=24000 + 60000=84000$.
  • At $(8000,6000)$: $R=8\times8000+10\times6000=64000+60000 = 124000$.

Answer:

The daily combination that maximizes revenue is 8000 litres of latex paint and 6000 litres of oil paint.