at an Oregan fiber manufacturing facility an analyst estimat

at an Oregan fiber manufacturing facility, an analyst estimates that the weekly number of pounds of acetate fibers that can be produced is given by the function:

z=f(x,y) = 1250ln(yx^2) +45(y^2+x)(x^3-2y)-(xy)^1/2

where z=the weekly # of pounds of acetate fiber produced

x= the # of skilled workers at the plant

y= the # of unskilled workers at the plant

a) on the second week of april, the production executive would like to see the effect of the jumber of unskilled workers towards the production performance. therefore, the number of skilled workers is fixed and the number of unskilled worker to be employed at the production line is depending on the situation. find the rate of change of the amount of acetate fiber produced in this week

Solution

As the number of skilled workers is fixed x is fixed and y will vary.

We have to calculate dz/dy here

d/dy(1250ln(yx^2) +45(y^2 +x)(x^3 -2y ) -(xy)^(1/2)

treating x as constant

1250d/dy(ln(yx^2)=1250/y by taking yx^2=u

d/dy(45(y^2 +x)(x^3 -2y) =45(2x^3y-2x-6y^2)

d/dy(xy)^(1/2)=(x)^(1/2)/2(y)^(1/2)

=1250/y+45(2x^3y-2x-6y^2)-((x)^(1/2))/(2(y)^(1/2)

Thank you