Question

Using x hours of skilled labor and y hours of unskilled labor, a manufacturer can produce Q(x,y)=40xy1/5 units each week. Currently 20 hours of skilled labor and 243 hours of unskilled labor are being used. Suppose the manufacturer reduces the skilled labor level by 2 hours and increases the unskilled labor by 3. Use calculus to determine the approximate effect of these changes on production.

Answer 1

I don't understand 40xy1/5.

dq=40 y1/5 dx + 40x1/5 dy

so if dx=-2, and dy=3
what is dq?

Answer 2

It's supposed to be Q(x,y)=40xy^(1/5)

Answer 3

First you find the derivative with respect to x and then to y.

With respect to x is 40y^(1/5)
With respect to y is (40/5)xy^(-4/5) or 8xy^(-4/5)

Each one of those will give us the marginal change per unit.

Marginal change per unit of x is 40y^(1/5) and marginal change per unit of y is 8xy^(-4/5). This problem is changing by more than one unit, so multiply each equation by how many units it is changing.

The change in x is -2 skilled labor hours so we get -80y^(1/5) and for +3 unskilled labor hours, we get 24xy^(-4/5).

Thus, the total change in production is 24xy^(-4/5)-80y^(1/5).

Once you plug in the original labor hours for x and y, you get 24(20)(243)^(-4/5)-80(243)^(1/5), which is the total change in production.