Question

suppose the labor cost (in dollars) for manufacturing a camera can be approximated by
$l(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y - 2xy + 112$
where $x$ 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.
labor cost will be minimized when $x = 10$ and $y = 12$.
the minimum labor cost is $\square$. (round to the nearest cent as needed.)

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 \(y = 12\)

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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 \(2\times10\times12 = 240\)

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Answer:

\(58\)