Question

A company produces two types of solar panels per year: x thousand of type A and y thousand from type B. The revenue and costs equations, in millions of dollars, for the year given as follows

R(x,y) = 5x+3y
c(x,y) = x2 -3xy+8y2 +11x-52y-3
Determine how many soalr panels of each type should be produced per year to maximize profit.
The company will achieve a maximum profit by selling (X) solar panels of type A and selling (X) solar panels of type B.

Answer 1

The profit is
`P(3,4) =  \$104 \  millon`

The number of solar panels of type a is 3 thousand

The number of solar panels of type B is 4 thousand

Explanation:

From the question we are told that

The revenue function is
`R(x,y) =  5x + 3y`

The cost function is
`c (x,y)  =  x^2 -3xy + 8y^2 + 11x -52y-3`

Generally the profit function is mathematically represented as

`P(x,y) =  R(x,y) -c(x,y) =  5x + 3y -x^2 +3xy -8y^2-11x+52y+3`

Now the next step is to differentiate the profit function partially

`P_x  =  (\delta P)/(\delta x ) =  -2x+3y-6`

`P_y  =  (\delta P)/(\delta y)  =  3x - 16y+56`

At maximum or minimum
`P_x  =  0` so

`-2x +3y-6 = 0 --- (1)`

and
`P_y  =  0`

So

`3x -16y +56 =  0 ---(2)`

Solving equation 1 and 2 simultaneously using substitution method

From 1

`x =  (-6+3y)/(2)`

substituting this to 2

`3[(-6+3y)/(2) ] -16y + 56 = 0`

multiply through by 2

`-18+ 9y - 32y + 112  = 0`

=>
`y  =  4`

So

`x =  (-6+3 (4))/(2) =  3`

So the critical point is (v,w) = (3, 4)

Now differentiating
`P_x` partially and substituting the critical point s we have

`P_(xx) |_(3,4)= -2`

Now differentiating
`P_y` partially and substituting the critical point s we have

`P_(yy) |_(3,4)= -16`

`P_(xy) |_(3,4)=  3`

Now to determine whether the obtained critical point is maximum or minimum the expression

`D  =  P_(xx)|_(3,4) * P_(yy)|_(3,4) - [P_(xy)|_(3,4) ]^2` must be greater than zero so

`(-2) *  (-16)- 3^2 = 23>0`

So \

The maximum price is mathematically evaluated as

`P(3,4) = 5(3) + 3(4) -(3)^2 +3(3)(4) -8(4)^2-11(3)+52(4)+3`

`P(3,4) =  \$104 \  millon`

So

The number of solar panels of type a is 3 thousand

The number of solar panels of type B is 4 thousand