Mathematics, 08.09.2020 02:01, ijohnh14

Suppose that the profit (in hundreds of dollars) of a certain firm is approximated by: P(x, y) = 1500 +36x - 1.5x^2 + 120y - 2y^2
where x is the cost of a unit of labor and y is the cost of a unit of goods. Find the values of x and y that mazimize profit. Find the maximum profit.

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answered: brooklyn4932

Step-by-step explanation:.

Given the profit (in hundreds of dollars) of a certain firm is approximated by:

P(x,y) = 1500 +36x - 1.5x^2 + 120y - 2y^2 where

x is the cost of a unit of labor

y is the cost of a unit of goods.

The maximum value of x and y occurs when dP/dx = 0 and dP/dy = 0

dP/dx is the differential of the function with respect to x taking y as a constant

dP/dy is the differential of the function with respect to y taking x as a constant

dP/dx = 0+36-2(1.5)x^{2-1}+0-0

dP/dx = 36-3x

If dP/dx = 0

36-3x = 0

36-3x-36 = 0-36

-3x = -36

x = -36/-3

x = 12

Similarly;

dP/dy = 0+0-0+120-2(2)y^{2-1}

dP/dy = 120-4y

Since dP/dy = 0

120-4y = 0

Subtract 120 from both sides

120-4y-120 = 0-120

-4y = -120

y = -120/-4

y = 30

Hence the values of x and y that maximize the profit are 12 and 30 respectively.

To find the maximum profit, we will substitute x = 12 and y = 30 into the given function

P(x,y) = 1500 +36x - 1.5x^2 + 120y - 2y^2

P(12,60) = 1500 +36(12) - 1.5(12)^2 + 120(30) - 2(30)^2

P(12,60) = 1500 +36(12) - 1.5(144) + 120(30) - 2(900)

P(12,60) = 1500 + 432 - 216 + 3600 - 1800

P(12,60) = 1932-216+1800

P(12,60) = 3,516

Since the profit is in hundreds if dollars,

P(12,60) = 3516×100

P(12,60) = $351,600

Hence the maximum profit is $351,600

answered: Guest

the system of inequity is

$y\leq -x+4$

$3y> x$

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

the solid line passing through the points (0,4) and (4,0).

$y-4=\frac{0-4}{4-0}(x-0)$

$y=-x+4$

the related equation of solid line is $y=-x+4$ .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

$y-0=\frac{0-1}{3-0}(x-0)$

$3y=x$

the related equation of dashed line is $3y=x$

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

$1=-0+4$

$1=4$

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be $\leq$ . the required inequality is