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.
(image of a graph with a purple region, a table with x, y, p values, 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: \( 9\times0 = 0 \) and \( 8\times0 = 0 \), so \( P = 0 + 0 = 0 \).

For the other corner points:

• When \( x = 6 \), \( y = 0 \):

Step1: Substitute x=6, y=0 into P

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

Step2: Calculate the value

\( 9\times6 = 54 \) and \( 8\times0 = 0 \), so \( P = 54 + 0 = 54 \).

• When \( x = 0 \), \( y = 7 \):

Step1: Substitute x=0, y=7 into P

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

Step2: Calculate the value

\( 9\times0 = 0 \) and \( 8\times7 = 56 \), so \( P = 0 + 56 = 56 \).

• When \( x = 3 \), \( y = 4 \):

Step1: Substitute x=3, y=4 into P

\( P = 9(3) + 8(4) \)

Step2: Calculate the value

\( 9\times3 = 27 \) and \( 8\times4 = 32 \), so \( P = 27 + 32 = 59 \).

Now, to find the maximum value of \( P \), we compare the values: \( 0 \), \( 54 \), \( 56 \), \( 59 \). The largest among them is \( 59 \).

Answer:

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