Questions: Find the maximum value of [ P=9 x+8 y ] subject to the following constraints: Now, find the coordinates of the corner points. x y 0 0 6 0 0 7 [?] 8 x+6 y leq 48 7 x+7 y leq 49 x geq 0 y geq 0

Find the maximum value of [ P=9 x+8 y ] subject to the following constraints: Now, find the coordinates of the corner points. x y 0 0 6 0 0 7 [?] 8 x+6 y leq 48 7 x+7 y leq 49 x geq 0 y geq 0
Transcript text: Find the maximum value of \[ P=9 x+8 y \] subject to the following constraints: Now, find the coordinates of the corner points. \begin{tabular}{ll} $x$ & $y$ \\ 0 & 0 \\ 6 & 0 \\ 0 & 7 \\ {$[?]$} \end{tabular}$\quad\left\{\begin{array}{l}8 x+6 y \leq 48 \\ 7 x+7 y \leq 49 \\ x \geq 0 \\ y \geq 0\end{array}\right.$
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Solution

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Solution Steps

Step 1: Find the intersection point of the two lines

The two lines are defined by the equations: $8x + 6y = 48$ $7x + 7y = 49$ which simplifies to $x + y = 7$ or $y = 7 - x$

Substitute $y$ in the first equation: $8x + 6(7 - x) = 48$ $8x + 42 - 6x = 48$ $2x = 6$ $x = 3$

Now substitute $x$ back into $y = 7 - x$: $y = 7 - 3$ $y = 4$

So the intersection point is $(3, 4)$.

Step 2: List the corner points

The corner points are $(0, 0)$, $(6, 0)$, $(0, 7)$ and $(3, 4)$.

Step 3: Evaluate P at each corner point

$P = 9x + 8y$

  • $(0, 0)$: $P = 9(0) + 8(0) = 0$
  • $(6, 0)$: $P = 9(6) + 8(0) = 54$
  • $(0, 7)$: $P = 9(0) + 8(7) = 56$
  • $(3, 4)$: $P = 9(3) + 8(4) = 27 + 32 = 59$

Final Answer:

The maximum value of P is 59. The coordinates of the missing corner point are $(3, 4)$.

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