optimization problems\nsummary 5: paint factory\na paint factory produces oil paint and latex paint. at…

optimization problems\nsummary 5: paint factory\na 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.\nx = number of l of latex paint

optimization problems\nsummary 5: paint factory\na 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.\nx = number of l of latex paint

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of litres (in thousands) of latex - paint and $y$ be the number of litres (in thousands) of oil - paint.

Step2: Set up constraints

We have the following constraints:

  1. $y\geq2x$ (At least 2 L of oil - paint is produced for every 1 L of latex - paint).
  2. $x\leq8$ (The factory can produce up to 8 thousand litres of latex - paint).
  3. $y\leq6$ (The factory can produce up to 6 thousand litres of oil - paint).
  4. $x\geq0,y\geq0$ (Non - negative production amounts).

Step3: Set up the objective function

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

Step4: Analyze the corner points

  1. Intersection of $y = 2x$ and $y = 6$: Substitute $y = 6$ into $y = 2x$, we get $6=2x$, so $x = 3$. The point is $(3,6)$.
  2. Intersection of $y = 2x$ and $x = 8$ is not valid since when $x = 8$, $y=16$ but $y\leq6$.
  3. Intersection of $x = 8$ and $y = 6$: The point is $(8,6)$ but it does not satisfy $y\geq2x$.
  4. Intersection of $y = 2x$ and $x = 0$ gives the point $(0,0)$ with revenue $R=0$.
  5. Intersection of $x = 0$ and $y = 6$ gives the point $(0,6)$ with revenue $R = 10\times6=60$.
  6. For the point $(3,6)$:
    • Substitute into the revenue function $R=8\times3 + 10\times6=24 + 60=84$.

Answer:

The factory should produce 3000 litres of latex paint and 6000 litres of oil paint to maximize the revenue.