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 $y$ be the number of liters of oil - paint. We know that $2\leq y\leq6000$ and $0\leq x\leq8000$. The revenue function $R$ is given by $R = 8x+10y$.

Step2: Consider the constraints

The constraints are $y\geq2x$ (at least 2 L of oil - paint is produced for every 1 L of latex - paint), $y\leq6000$, and $x\leq8000$.

Step3: Analyze the objective function

We want to maximize $R = 8x + 10y$. From $y\geq2x$, we can rewrite the revenue function in terms of one variable when considering the boundary conditions. If we consider the boundary $y = 2x$. Substitute $y = 2x$ into $R$: $R=8x + 10(2x)=8x+20x=28x$. Since $x\leq8000$, when $x = 8000$, then $y = 2x=16000$, but $y\leq6000$. So we consider the intersection of $y = 6000$ and $y = 2x$. Solving $2x=6000$ gives $x = 3000$.

Step4: Calculate the revenue

Substitute $x = 3000$ and $y = 6000$ into the revenue function $R=8x + 10y$. Then $R=8\times3000+10\times6000=24000 + 60000=84000$.

Answer:

The factory should produce 3000 liters of latex - paint and 6000 liters of oil - paint to maximize the revenue, and the maximum revenue is $84000$.