suppose the labor cost (in dollars) for manufacturing a camera can be approximated by\n l(x,y)=\frac{3}{2}x^{…

suppose the labor cost (in dollars) for manufacturing a camera can be approximated by\n l(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y - 2xy + 112 \nwhere ( x ) is the number of hours required by a skilled craftsperson and ( y ) is the number of hours required by a semiskilled person. find values of ( x ) and ( y ) that minimize the labor cost. find the minimum labor\ncost.\nlabor cost will be minimized when ( x=square ) and ( y=square ).

suppose the labor cost (in dollars) for manufacturing a camera can be approximated by\n l(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y - 2xy + 112 \nwhere ( x ) is the number of hours required by a skilled craftsperson and ( y ) is the number of hours required by a semiskilled person. find values of ( x ) and ( y ) that minimize the labor cost. find the minimum labor\ncost.\nlabor cost will be minimized when ( x=square ) and ( y=square ).

Answer

Explanation:

Step1: Find the partial derivatives

Find (L_x) and (L_y). For (L(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y-2xy + 112), (L_x = 3x-6 - 2y) (using power rule ((x^n)^\prime=nx^{n - 1})) (L_y=2y-4 - 2x)

Step2: Set the partial derivatives equal to zero

Set (L_x = 0) and (L_y = 0). (\begin{cases}3x-2y-6 = 0\-2x + 2y-4=0\end{cases}) Add the two equations: ((3x-2y-6)+(-2x + 2y-4)=0), which simplifies to (x-10 = 0), so (x = 10) Substitute (x = 10) into (3x-2y-6 = 0), we get (3\times10-2y-6 = 0), (30-2y-6=0), (24-2y = 0), (y = 12)

Step3: Check the second - derivative conditions

Find (L_{xx}=3), (L_{xy}=-2), (L_{yy}=2) The discriminant (D = L_{xx}L_{yy}-(L_{xy})^{2}=3\times2-(-2)^{2}=6 - 4=2>0) and (L_{xx}=3>0)

Answer:

(x = 10) and (y = 12)