5\nselect the correct location on the coordinate plane.\nruhana owns a workshop where her team of…

5\nselect the correct location on the coordinate plane.\nruhana owns a workshop where her team of technicians refurbishes tv sets and dvd players, and then she sells them for a profit. she has a weekly sales target of at least 35 tv sets or dvd players. additionally, she must ensure that her team collectively works at least 100 hours each week. it takes 4 hours to refurbish a tv set and 2 hours to refurbish a dvd player.\nit costs $75 to refurbish a tv set and $40 to refurbish a dvd player. if ruhana wants to minimize costs, which point represents the optimal number of tv sets and dvd players that her team should refurbish each week?

5\nselect the correct location on the coordinate plane.\nruhana owns a workshop where her team of technicians refurbishes tv sets and dvd players, and then she sells them for a profit. she has a weekly sales target of at least 35 tv sets or dvd players. additionally, she must ensure that her team collectively works at least 100 hours each week. it takes 4 hours to refurbish a tv set and 2 hours to refurbish a dvd player.\nit costs $75 to refurbish a tv set and $40 to refurbish a dvd player. if ruhana wants to minimize costs, which point represents the optimal number of tv sets and dvd players that her team should refurbish each week?

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of TV - sets and $y$ be the number of DVD - players.

Step2: Set up constraints

  1. Sales target constraint: $x + y\geq35$.
  2. Working - hours constraint: $4x + 2y\geq100$, which simplifies to $2x + y\geq50$. Also, $x\geq0,y\geq0$ since the number of items cannot be negative.

Step3: Define the cost function

The cost function $C=75x + 40y$.

Step4: Rewrite constraints in slope - intercept form

The first constraint $y\geq - x + 35$. The second constraint $y\geq - 2x+50$.

Step5: Find the intersection points of the boundary lines

Intersection of $y=-x + 35$ and $y=-2x + 50$: Set $-x + 35=-2x + 50$. Add $2x$ to both sides: $2x - x+35=50$. $x + 35=50$, so $x = 15$. Substitute $x = 15$ into $y=-x + 35$, we get $y=-15 + 35=20$. The intersection point of $y=-x + 35$ and $x = 0$ is $(0,35)$. The intersection point of $y=-2x + 50$ and $x = 0$ is $(0,50)$. The intersection point of $y=-2x + 50$ and $y = 0$ is $2x=50$, so $x = 25$ and the point is $(25,0)$.

Step6: Evaluate the cost function at the corner - points

  1. At $(0,50)$: $C=75\times0+40\times50=2000$.
  2. At $(15,20)$: $C=75\times15 + 40\times20=1125+800=1925$.
  3. At $(25,0)$: $C=75\times25+40\times0=1875$.
  4. At $(0,35)$: $C=75\times0+40\times35 = 1400$. But $(0,35)$ does not satisfy $2x + y\geq50$ ($2\times0+35=35\lt50$).

Answer:

The point $(25,0)$ represents the optimal number of TV - sets and DVD - players that her team should refurbish each week to minimize costs.