Maximizing Profit The total weeldy revenue (in dollars) of the Country Workshop realized in manufacturing and selling its rolitop desks is given by the following equation, where $x$ denotes the number of finished units and $y$ denotes the number of unfinished units manufactured and sold each week.
\[
R(x, y)=-0.25 x^{2}-0.2 y^{2}-0.2 x y+200 x+150 y
\]

The totat weebly cost attributable to the manufacture of these desis is given by the following equation, where $C(x, y)$ is in dollars.
\[
C(x, n)=90 x+90 y+1300
\]

Determine how many finished units and how many unfinished units the company should manufacture each week in order to maximize its profit.
finished
90
$\mathbf{X}$ units
unfinished
$\mathbf{X}$ units

What is the maximum profit reallzabie?
s
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Step 1 ：\( P(x, y) = R(x, y) - C(x, y) \)

Step 2 ：\( P(x, y) = (-0.25x^2 - 0.2y^2 - 0.2xy + 200x + 150y) - (90x + 90y + 1300) \)

Step 3 ：\( P(x, y) = -0.25x^2 - 0.2y^2 - 0.2xy + 200x + 150y - 90x - 90y - 1300 \)

Step 4 ：\( P(x, y) = -0.25x^2 - 0.2y^2 - 0.2xy + 110x + 60y - 1300 \)

Step 5 ：\( \frac{\partial P}{\partial x} = -0.5x - 0.2y + 110 \)

Step 6 ：\( \frac{\partial P}{\partial y} = -0.4y - 0.2x + 60 \)

Step 7 ：\( -0.5x - 0.2y + 110 = 0 \)

Step 8 ：\( -0.4y - 0.2x + 60 = 0 \)

Step 9 ：\( -y - 0.5x + 150 = 0 \)

Step 10 ：\( -0.5x - y + 110 = 0 \)

Step 11 ：\( -y - 0.5x + 150 = 0 \)

Step 12 ：\( -1.5x + 260 = 0 \)

Step 13 ：\( x = \frac{260}{1.5} \)

Step 14 ：\( x = 173.\overline{3} \)

Step 15 ：\( x = 173 \)

Step 16 ：\( -0.4y - 0.2(173) + 60 = 0 \)

Step 17 ：\( -0.4y - 34.6 + 60 = 0 \)

Step 18 ：\( -0.4y + 25.4 = 0 \)

Step 19 ：\( y = \frac{25.4}{0.4} \)

Step 20 ：\( y = 63.5 \)

Step 21 ：\( y = 63 \)

Step 22 ：\( \boxed{x = 173, y = 63} \)