A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thous

Burhan Hopper 2021-01-05 Answered
A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the $25 shirt is given by
C(x,y)=4x26xy+3y2+20x+19y12.
How many of each type of shirt must be produced and sold in order to maximize profit?
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Expert Answer

Bentley Leach
Answered 2021-01-06 Author has 109 answers

Revenue function
R(x,y)=18x+25y
Profit function
C(x,y)=R(x,y)C(x,y)
C(x,y)=18x+25y)(4x26xy+3y2+20x+19y12)
C(x,y)=4x2+6×y3y22x+6y+12
First we will find the critical point. Now,
Px(x,y)=8x+6y2
Py(x,y)=6x6y+6
Px(x,y)=8
Pyy(x,y)=6
Py(x,y)=0
6x-6y+6=0
y=x+1
And Px(x,y)=0
6y-8x-2=0
6(x+1)-8x=2
x=2 (i)
So, y=x+1
y=2 (ii)
Hence, the critical point is (2,3)
Since fx,fyy,andfxy all are constant so we will need to pkug critical point to function
D(x,y)=fxfyyfxy2
=(8)(6)(6)2
=12>0
Since, D>0, and fx<0
The critical point P(2,3) has a relative maximum.
And the maximum value of profit function
P(2,3)=4(2)2+6233(3)222+63+12
P(2,3)=19
The maximum profit of $19000 will be earned if 2000 shirt of $18 and 3000 shirts of $25 are produced and sold.

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