an electronics company assembles two types of tvs: plasma and lcd. a plasma tv costs $400 to assemble and…

an electronics company assembles two types of tvs: plasma and lcd. a plasma tv costs $400 to assemble and takes 40 hours of labor. the lcd tv costs $250 and requires 30 hours of labor. the company has $20 000 in capital, and 2160 hours of labor available for assembly. what is the maximum number of tvs the electronics company can assemble?

an electronics company assembles two types of tvs: plasma and lcd. a plasma tv costs $400 to assemble and takes 40 hours of labor. the lcd tv costs $250 and requires 30 hours of labor. the company has $20 000 in capital, and 2160 hours of labor available for assembly. what is the maximum number of tvs the electronics company can assemble?

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of plasma - TVs and $y$ be the number of LCD - TVs.

Step2: Set up constraints

The cost constraint: $400x + 250y\leq20000$ (capital constraint), which simplifies to $8x + 5y\leq400$. The labor - hour constraint: $30x+40y\leq2160$, which simplifies to $3x + 4y\leq216$. Also, $x\geq0,y\geq0$ (non - negativity constraints).

Step3: Rewrite inequalities for graphing

For $8x + 5y=400$, when $x = 0$, $y = 80$; when $y = 0$, $x = 50$. For $3x + 4y=216$, when $x = 0$, $y = 54$; when $y = 0$, $x = 72$.

Step4: Find the feasible region

The feasible region is bounded by the $x$ - axis, $y$ - axis, $8x + 5y = 400$ and $3x + 4y = 216$.

Step5: Find the corner points of the feasible region

  1. Intersection of $x = 0$ and $y = 0$: $(0,0)$
  2. Intersection of $x = 0$ and $3x + 4y=216$: Substitute $x = 0$ into $3x + 4y=216$, we get $y = 54$. So the point is $(0,54)$.
  3. Intersection of $y = 0$ and $8x + 5y=400$: Substitute $y = 0$ into $8x + 5y=400$, we get $x = 50$. So the point is $(50,0)$.
  4. Solve the system of equations $\begin{cases}8x + 5y=400\3x + 4y=216\end{cases}$ Multiply the first equation by 4 and the second equation by 5: $\begin{cases}32x+20y = 1600\15x + 20y=1080\end{cases}$ Subtract the second equation from the first: $32x+20y-(15x + 20y)=1600 - 1080$ $32x+20y - 15x-20y=520$ $17x=520$ $x=\frac{520}{17}\approx30.59$ Substitute $x=\frac{520}{17}$ into $8x + 5y=400$: $8\times\frac{520}{17}+5y=400$ $\frac{4160}{17}+5y=400$ $5y=400-\frac{4160}{17}$ $5y=\frac{6800 - 4160}{17}$ $5y=\frac{2640}{17}$ $y=\frac{528}{17}\approx31.06$

Step6: Define the objective function

The objective function is $N=x + y$ (total number of TVs). Evaluate the objective function at the corner points:

  • At $(0,0)$: $N=0+0 = 0$
  • At $(0,54)$: $N=0 + 54=54$
  • At $(50,0)$: $N=50+0 = 50$
  • At $(\frac{520}{17},\frac{528}{17})$: $N=\frac{520}{17}+\frac{528}{17}=\frac{1048}{17}\approx61.65$

Since we are dealing with whole number of TVs, we can also test integer points within the feasible region close to the non - integer corner point. By checking integer points, we find that when $x = 30$ and $y = 31$: For the capital constraint: $400\times30+250\times31=12000 + 7750=19750\leq20000$ For the labor - hour constraint: $30\times30+40\times31=900+1240 = 2140\leq2160$ $N=x + y=30 + 31=61$

Answer:

61