# 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

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
$C\left(x,y\right)=4{x}^{2}-6xy+3{y}^{2}+20x+19y-12.$
How many of each type of shirt must be produced and sold in order to maximize profit?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Bentley Leach

Revenue function
$R\left(x,y\right)=18x+25y$
Profit function
$C\left(x,y\right)=R\left(x,y\right)-C\left(x,y\right)$
$C\left(x,y\right)=\cdot 18x+25y\right)-\left(4{x}^{2}-6xy+3{y}^{2}+20x+19y-12\right)$
$C\left(x,y\right)=-4{x}^{2}+6×y-3{y}^{2}-2x+6y+12$
First we will find the critical point. Now,
${P}_{x}\left(x,y\right)=-8x+6y-2$
${P}_{y}\left(x,y\right)=6x-6y+6$
${P}_{x}\left(x,y\right)=-8$
${P}_{yy}\left(x,y\right)=-6$
${P}_{y}\left(x,y\right)=0$
6x-6y+6=0
y=x+1
And ${P}_{x}\left(x,y\right)=0$
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 ${f}_{x},{f}_{yy},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{xy}$ all are constant so we will need to pkug critical point to function
$D\left(x,y\right)={f}_{x}{f}_{yy}-{f}_{xy}^{2}$
$=\left(-8\right)\left(-6\right)-{\left(-6\right)}^{2}$
=12>0
Since, D>0, and ${f}_{x}<0$
The critical point P(2,3) has a relative maximum.
And the maximum value of profit function
$P\left(2,3\right)=-4{\left(2\right)}^{2}+6\cdot 2\cdot 3-3{\left(3\right)}^{2}-2\cdot 2+6\cdot 3+12$
$P\left(2,3\right)=19$
The maximum profit of $19000 will be earned if 2000 shirt of$18 and 3000 shirts of \$25 are produced and sold.