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 cost.\nlabor cost will be minimized when $x = 10$ and $y = 12$.\nthe minimum labor cost is $\\square$. (round to the nearest cent as needed.)

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 cost.\nlabor cost will be minimized when $x = 10$ and $y = 12$.\nthe minimum labor cost is $\\square$. (round to the nearest cent as needed.)

Answer

Explanation:

Step1: Substitute (x = 10) and (y = 12) into (L(x,y))

Given (L(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y-2xy + 112) Substitute (x = 10) and (y = 12) [ \begin{align*} L(10,12)&=\frac{3}{2}\times(10)^{2}+(12)^{2}-6\times10-4\times12-2\times10\times12 + 112\ \end{align*} ]

Step2: Calculate each term

  • Calculate (\frac{3}{2}\times(10)^{2}=\frac{3}{2}\times100 = 150)
  • Calculate ((12)^{2}=144)
  • Calculate (6\times10 = 60)
  • Calculate (4\times12=48)
  • Calculate (2\times10\times12 = 240) [ \begin{align*} L(10,12)&=150+144-60 - 48-240+112\ &=(150 + 144+112)-(60 + 48+240)\ &=406 - 348\ &=58 \end{align*} ]

Answer:

(58)