ruhana owns a workshop where her team of technicians refurbishes tv sets and dvd players, and then she sells…

ruhana 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. it 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?

ruhana 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. it 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, $x\geq0,y\geq0$.

Step3: Define the cost function

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

Step4: Analyze the corner - points

We need to check the cost at each of the corner - points of the feasible region (the pink points on the graph). Let's assume the corner - points are $(x_1,y_1),(x_2,y_2),\cdots$. Calculate the cost $C_i=75x_i + 40y_i$ for each corner - point $(x_i,y_i)$. For example, if one corner - point is $(0, 50)$: $C_1=75\times0 + 40\times50=2000$. If another corner - point is $(15,20)$: $C_2=75\times15 + 40\times20=1125+800 = 1925$. If a corner - point is $(25,0)$: $C_3=75\times25+40\times0 = 1875$.

Answer:

The point that minimizes the cost is the one with the lowest cost value among the corner - points. Assuming the calculations above, if the lowest cost is at the point $(25,0)$, then the optimal number of TV sets is 25 and the optimal number of DVD players is 0. (The actual optimal point depends on the exact coordinates of the corner - points on the graph, but the method of calculating the cost at each corner - point is as shown above).