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.

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.

Answer 4

To determine the approximate effect of the changes on production, we need to find the partial derivatives of the production function with respect to skilled labor (x) and unskilled labor (y), and then calculate the effects of the changes in labor levels.

Let's start by finding the partial derivative of the production function Q(x, y) with respect to skilled labor (x):

∂Q/∂x = ∂(40xy^1/5)/∂x

To find this derivative, we treat y as a constant and differentiate the function with respect to x:

∂Q/∂x = 40y^1/5 * ∂(x)/∂x

Since x appears linearly, the derivative of x with respect to x is simply 1:

∂Q/∂x = 40y^1/5

Next, let's find the partial derivative of the production function with respect to unskilled labor (y):

∂Q/∂y = ∂(40xy^1/5)/∂y

Again, treating x as a constant, we differentiate the function with respect to y:

∂Q/∂y = (1/5) * 40x * y^(-4/5)

Simplifying further:

∂Q/∂y = 8xy^(-4/5)

Now that we have the partial derivatives, we can calculate the effects of the changes in labor levels. The manufacturer is reducing skilled labor by 2 hours, so we substitute x = 20 - 2 = 18 into the partial derivatives:

∂Q/∂x = 40y^1/5 = 40(243)^(1/5) ≈ 146.15

∂Q/∂y = 8xy^(-4/5) = 8(18)(243)^(-4/5) ≈ 0.128

These values represent the rates of change of production with respect to skilled labor and unskilled labor, respectively. However, to find the approximate effect of the changes on production, we need to multiply these rates of change by the changes in labor levels.

The manufacturer is also increasing unskilled labor by 3 hours, so we substitute y = 243 + 3 = 246 into the partial derivatives:

∂Q/∂x = 40y^1/5 = 40(246)^(1/5) ≈ 147.02

∂Q/∂y = 8xy^(-4/5) = 8(18)(246)^(-4/5) ≈ 0.1287

Now we have the approximate effects of the changes in labor levels:

The reduction in skilled labor by 2 hours is estimated to decrease production by approximately 146.15 units per week.

The increase in unskilled labor by 3 hours is estimated to increase production by approximately 0.1287 units per week.

Note that these are approximate effects based on the partial derivatives calculated at specific labor levels. The actual effects on production may differ slightly due to the assumptions made in this calculation.