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
$X$ units
unfinished
$X$ units

What is the maximum profit reallzabie?
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Answer

$x = 173, y = 63$

Steps

Step 1 ： $P (x, y) = R (x, y) - C (x, y)$

Step 2 ： $P (x, y) = (- 0.25 x^{2} - 0.2 y^{2} - 0.2 x y + 200 x + 150 y) - (90 x + 90 y + 1300)$

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

Step 4 ： $P (x, y) = - 0.25 x^{2} - 0.2 y^{2} - 0.2 x y + 110 x + 60 y - 1300$

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

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

Step 7 ： $- 0.5 x - 0.2 y + 110 = 0$

Step 8 ： $- 0.4 y - 0.2 x + 60 = 0$

Step 9 ： $- y - 0.5 x + 150 = 0$

Step 10 ： $- 0.5 x - y + 110 = 0$

Step 11 ： $- y - 0.5 x + 150 = 0$

Step 12 ： $- 1.5 x + 260 = 0$

Step 13 ： $x = \frac{260}{1.5}$

Step 14 ： $x = 173. \overset{―}{3}$

Step 15 ： $x = 173$

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

Step 17 ： $- 0.4 y - 34.6 + 60 = 0$

Step 18 ： $- 0.4 y + 25.4 = 0$

Step 19 ： $y = \frac{25.4}{0.4}$

Step 20 ： $y = 63.5$

Step 21 ： $y = 63$

Step 22 ： $x = 173, y = 63$