Question

find the maximum value of
p = 9x + 8y
subject to the following constraints:
now find the profit, p, at each corner point using the
above function.
(the image shows a graph with a purple region, a table with x, y, p values (0,0 with p unknown, 6,0; 0,7; 3,4), and constraints: 8x + 6y ≤ 48, 7x + 7y ≤ 49, x ≥ 0, y ≥ 0)

Explanation:

Step1: Substitute x=0, y=0 into P

We have the profit function \( P = 9x + 8y \). Substitute \( x = 0 \) and \( y = 0 \) into this function.
\( P = 9(0) + 8(0) \)

Step2: Calculate the value

Simplify the expression: \( 9\times0 + 8\times0 = 0 + 0 = 0 \)

Now we will find the values for the other corner points as well to determine the maximum.

Step3: Substitute x=6, y=0 into P

\( P = 9(6) + 8(0) \)
\( P = 54 + 0 = 54 \)

Step4: Substitute x=0, y=7 into P

\( P = 9(0) + 8(7) \)
\( P = 0 + 56 = 56 \)

Step5: Substitute x=3, y=4 into P

\( P = 9(3) + 8(4) \)
\( P = 27 + 32 = 59 \)

Now we compare the values: 0, 54, 56, 59. The largest value among them is 59.

Answer:

The maximum value of \( P \) is 59. The values at each corner point are: (0,0) - 0, (6,0) - 54, (0,7) - 56, (3,4) - 59.