19. a small clothing company makes two types of jeans - straight leg and flare cut. straight leg jeans…

19. a small clothing company makes two types of jeans - straight leg and flare cut. straight leg jeans require 10 minutes of cutting time and 15 minutes of sewing time. flare cut jeans require 20 minutes of cutting time and 10 minutes of sewing time. the company has a maximum of eight hours of cutting time and 10 hours of sewing time. every pair of straight leg jeans made earns a profit of $20 and every pair of flare cut jeans made earns a profit of $25. how many of each type of jeans should be made for maximum profit? state the maximum profit.

19. a small clothing company makes two types of jeans - straight leg and flare cut. straight leg jeans require 10 minutes of cutting time and 15 minutes of sewing time. flare cut jeans require 20 minutes of cutting time and 10 minutes of sewing time. the company has a maximum of eight hours of cutting time and 10 hours of sewing time. every pair of straight leg jeans made earns a profit of $20 and every pair of flare cut jeans made earns a profit of $25. how many of each type of jeans should be made for maximum profit? state the maximum profit.

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of straight - leg jeans and $y$ be the number of flare - cut jeans.

Step2: Set up constraints

Cutting time constraint: $10x + 20y\leqslant8\times60$ (since 8 hours = 480 minutes), simplifies to $x + 2y\leqslant48$. Sewing time constraint: $15x+10y\leqslant10\times60$ (since 10 hours = 600 minutes), simplifies to $3x + 2y\leqslant120$. Also, $x\geqslant0,y\geqslant0$ as the number of jeans cannot be negative.

Step3: Set up the objective function

The profit function $P = 20x+25y$.

Step4: Find the corner points

Solve the system of equations $\begin{cases}x + 2y=48\3x + 2y=120\end{cases}$. Subtract the first equation from the second: $(3x + 2y)-(x + 2y)=120 - 48$, $2x=72$, $x = 36$. Substitute $x = 36$ into $x + 2y=48$, we get $36+2y=48$, $2y=12$, $y = 6$. The corner points of the feasible region are $(0,0),(0,24),(40,0),(36,6)$.

Step5: Evaluate the objective function at corner points

  • At $(0,0)$: $P=20\times0 + 25\times0=0$.
  • At $(0,24)$: $P=20\times0+25\times24 = 600$.
  • At $(40,0)$: $P=20\times40+25\times0=800$.
  • At $(36,6)$: $P=20\times36+25\times6=720 + 150=870$.

Answer:

36 straight - leg jeans and 6 flare - cut jeans should be made for a maximum profit of $870.