what is the maximum value of $p = 24x + 30y$, given the constraints on $x$ and $y$ listed below?\n$x + y…

what is the maximum value of $p = 24x + 30y$, given the constraints on $x$ and $y$ listed below?\n$x + y \\leq 5$\n$x - y \\geq -1$\n$x \\geq 0$\n$y \\geq 0$\n\\bigcirc\\ 120\n\\bigcirc\\ 132\n\\bigcirc\\ 138\n\\bigcirc\\ 150

what is the maximum value of $p = 24x + 30y$, given the constraints on $x$ and $y$ listed below?\n$x + y \\leq 5$\n$x - y \\geq -1$\n$x \\geq 0$\n$y \\geq 0$\n\\bigcirc\\ 120\n\\bigcirc\\ 132\n\\bigcirc\\ 138\n\\bigcirc\\ 150

Answer

Explanation:

Step1: Find feasible region vertices

The feasible region is bounded by the intersection of constraints. We solve pairs of equations to find vertices:

  1. Intersection of $x=0$ and $x+y=5$: $x=0, y=5$
  2. Intersection of $x+y=5$ and $x-y=-1$: Solve $\begin{cases}x+y=5 \ x-y=-1\end{cases}$, add equations: $2x=4 \implies x=2$, then $y=3$
  3. Intersection of $x-y=-1$ and $y=0$: $x=-1$, but $x\geq0$, so use intersection of $x-y=-1$ and $x=0$: $x=0, y=1$ (corrected, since $x\geq0$)
  4. Intersection of $x=0$ and $y=0$: $(0,0)$
  5. Intersection of $y=0$ and $x+y=5$: $(5,0)$

Valid vertices (within all constraints): $(0,0)$, $(5,0)$, $(2,3)$, $(0,5)$, $(0,1)$

Step2: Evaluate $P$ at each vertex

  • At $(0,0)$: $P=24(0)+30(0)=0$
  • At $(5,0)$: $P=24(5)+30(0)=120$
  • At $(2,3)$: $P=24(2)+30(3)=48+90=138$
  • At $(0,[LLM SSE On Failure]