Answer:
The profit is [tex]P(3,4) = \$104 \ millon[/tex]
The number of solar panels of type a is 3 thousand
The number of solar panels of type B is 4 thousand
Step-by-step explanation:
From the question we are told that
The revenue function is [tex]R(x,y) = 5x + 3y[/tex]
The cost function is [tex]c (x,y) = x^2 -3xy + 8y^2 + 11x -52y-3[/tex]
Generally the profit function is mathematically represented as
[tex]P(x,y) = R(x,y) -c(x,y) = 5x + 3y -x^2 +3xy -8y^2-11x+52y+3[/tex]
Now the next step is to differentiate the profit function partially
[tex]P_x = \frac{\delta P}{\delta x } = -2x+3y-6[/tex]
[tex]P_y = \frac{\delta P}{\delta y} = 3x - 16y+56[/tex]
At maximum or minimum [tex]P_x = 0[/tex] so
[tex]-2x +3y-6 = 0 --- (1)[/tex]
and [tex]P_y = 0[/tex]
So
[tex]3x -16y +56 = 0 ---(2)[/tex]
Solving equation 1 and 2 simultaneously using substitution method
From 1
[tex]x = \frac{-6+3y}{2}[/tex]
substituting this to 2
[tex]3[\frac{-6+3y}{2} ] -16y + 56 = 0[/tex]
multiply through by 2
[tex]-18+ 9y - 32y + 112 = 0[/tex]
=> [tex]y = 4[/tex]
So
[tex]x = \frac{-6+3 (4)}{2} = 3[/tex]
So the critical point is (v,w) = (3, 4)
Now differentiating [tex]P_x[/tex] partially and substituting the critical point s we have
[tex]P_{xx} |_{3,4}= -2[/tex]
Now differentiating [tex]P_y[/tex] partially and substituting the critical point s we have
[tex]P_{yy} |_{3,4}= -16[/tex]
[tex]P_{xy} |_{3,4}= 3[/tex]
Now to determine whether the obtained critical point is maximum or minimum the expression
[tex]D = P_{xx}|_{3,4} * P_{yy}|_{3,4} - [P_{xy}|_{3,4} ]^2[/tex] must be greater than zero so
[tex](-2) * (-16)- 3^2 = 23>0[/tex]
So \
The maximum price is mathematically evaluated as
[tex]P(3,4) = 5(3) + 3(4) -(3)^2 +3(3)(4) -8(4)^2-11(3)+52(4)+3[/tex]
[tex]P(3,4) = \$104 \ millon[/tex]
So
The number of solar panels of type a is 3 thousand
The number of solar panels of type B is 4 thousand
