Suppose the labor cost in dollars for manufacturing a camera...

Suppose the labor cost (in dollars) for manufacturing a camera can be approximated by L(x,y) = 32x^2 + y^2 - 7x - 8y - 2xy + 197 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 = nothing and y = nothing. The minimum labor cost is $nothing. (Round to the nearest cent)

Transcript

00:01 Hello, let's have a look at the question.

00:04 So here it is given that lxy is equals to 32x squared plus y square minus 7x minus 8y minus twice xy and plus 1 97.

00:20 Now let us this be the first equation.

00:23 Now here x is the number of hours required by a skill craft person.

00:33 And y is the number of hours required by a semi -skilled person.

00:42 Now, we will first find the critical points with the help of first order partial derivative from 1.

00:48 So we will have that dl by dx is equal to 64x minus 7 minus 2y and the value for dl by dy will be equals to 2y minus 2x.

01:03 Now this will lead to a system of equation which will be as 64x minus twice of y equals to 7 and twice of y minus twice of x is equals to 8.

01:17 Now this is the second equation and this is the third equation.

01:21 Now let us solve these both equation here.

01:23 So we will add second and third we will get 64x minus twice of y equals to 7 and minus 2x plus twice of 1.

01:33 Y is equals to 8.

01:35 So here this will get cancelled and we will get 62 x is equal to 15.

01:41 So the value of x is equals to 15 divided by 602.

01:45 Now here we can find the value for y by putting x equals to 15 by 62 in the second equation.

01:55 So we will get 64x minus 2y is equals to 7 so now 64 multiplied by 15 by 62 minus twice of y is equal to 7 so we will get that twice of y is equal to 960 by 62 and minus 7 so from here y is equals to 263 divided by now here we got that the critical points are 15 by 62 and 263 by 62...

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