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

select 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 set 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?

select 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 set 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

The sales - target constraint: $x + y\geq35$. The labor - hour constraint: $4x + 2y\geq100$, which simplifies to $2x + y\geq50$. Also, $x\geq0$ and $y\geq0$.

Step3: Define the cost function

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

Step4: Find the corner - points of the feasible region

Solve the system of equations $\begin{cases}x + y=35\2x + y=50\end{cases}$. Subtract the first equation from the second: $(2x + y)-(x + y)=50 - 35$, so $x = 15$. Substitute $x = 15$ into $x + y=35$ gives $y=20$. The intersection of $x=0$ and $x + y=35$ gives the point $(0,35)$. The intersection of $y = 0$ and $2x + y=50$ gives the point $(25,0)$.

Step5: Evaluate the cost function at corner - points

For $(0,35)$: $C=75\times0 + 40\times35=1400$. For $(15,20)$: $C=75\times15+40\times20=1125 + 800=1925$. For $(25,0)$: $C=75\times25+40\times0 = 1875$.

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.