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

Explanation:

Step1: Define variables

Let $x$ be the number of plasma - TVs and $y$ be the number of LCD - TVs. The cost constraint is $400x + 250y\leq20000$, which simplifies to $8x + 5y\leq400$. The labor - hour constraint is $30x+40y\leq2160$, which simplifies to $3x + 4y\leq216$. Also, $x\geq0,y\geq0$ and $x,y$ are non - negative integers.

Step2: Express one variable from the cost equation

From $8x + 5y=400$, we get $y = 80-\frac{8}{5}x$. From $3x + 4y=216$, we get $y=\frac{216 - 3x}{4}$.

Step3: Solve the system of inequalities by finding the corner - points

First, consider the intersection of $8x + 5y=400$ and $3x + 4y=216$. Multiply the first equation by 4 and the second by 5: $32x+20y = 1600$ and $15x + 20y=1080$. Subtract the second equation from the first: $32x+20y-(15x + 20y)=1600 - 1080$. $32x+20y - 15x-20y=520$. $17x=520$, $x=\frac{520}{17}\approx30.59$. $y=\frac{216 - 3\times\frac{520}{17}}{4}=\frac{216\times17- 1560}{68}=\frac{3672-1560}{68}=\frac{2112}{68}\approx31.06$. Intersection of $8x + 5y = 400$ with $x = 0$ gives $y = 80$ (but this may not satisfy the labor - hour constraint). Intersection of $8x + 5y = 400$ with $y = 0$ gives $x = 50$. Intersection of $3x + 4y=216$ with $x = 0$ gives $y = 54$. Intersection of $3x + 4y=216$ with $y = 0$ gives $x = 72$. We also need to check non - negative integer solutions. From $8x + 5y\leq400$ and $3x + 4y\leq216$. Let's rewrite the inequalities as $y\leq80-\frac{8}{5}x$ and $y\leq\frac{216 - 3x}{4}$. We can test integer values of $x$. If $x = 0$, from the labor - hour constraint $y=\frac{216}{4}=54$ and from the cost constraint $y = 80$. So $y = 54$. If $x = 10$, then from the cost constraint $y=80 - 16 = 64$, from the labor - hour constraint $y=\frac{216-30}{4}=\frac{186}{4}=46.5$. So $y = 46$. If $x = 20$, then from the cost constraint $y=80 - 32 = 48$, from the labor - hour constraint $y=\frac{216 - 60}{4}=\frac{156}{4}=39$. If $x = 30$, then from the cost constraint $y=80 - 48 = 32$, from the labor - hour constraint $y=\frac{216 - 90}{4}=\frac{126}{4}=31.5$. So $y = 31$. If $x = 40$, then from the cost constraint $y=80 - 64 = 16$, from the labor - hour constraint $y=\frac{216 - 120}{4}=\frac{96}{4}=24$. If $x = 50$, then from the cost constraint $y=0$, and from the labor - hour constraint $y=\frac{216-150}{4}=\frac{66}{4}=16.5$. The total number of TVs $N=x + y$. We calculate $N$ for each valid pair $(x,y)$: For $(0,54)$, $N=54$. For $(10,46)$, $N=56$. For $(20,39)$, $N=59$. For $(30,31)$, $N=61$. For $(40,24)$, $N=64$. For $(50,0)$, $N=50$.

Answer:

64