MATH SOLVE

2 months ago

Q:
# CHAPTER SUMMARY647Using x skilled and y anskilled workers, a manufacturer can produce Q(x, y) = 60x^1/3y^2/3 units per day.Currently the manufacturer employs 10 skilledworkers and 40 unskilled workers and is planningto hire 1 additional skilled worker. Use calculus toestimate the corresponding change that themanufacturer should make in the level of unskilledlabor so that the total output will remain theER SUMMARYsame.

Accepted Solution

A:

Answer:The manufacturer should decrease the level of unskilled labor by 2 for the total output to stay the same.Step-by-step explanation:Q(x,y) = 60 x^β
y^β
Take the partial derivatives with respect to x and y.βQ/βx = 20 x^-β
y^β
βQ/βy = 40 x^β
y^-β
So the total differential is:dQ = βQ/βx dx + βQ/βy dydQ = 20 x^-β
y^β
dx + 40 x^β
y^-β
dyIf dQ = 0:0 = 20 x^-β
y^β
dx + 40 x^β
y^-β
dyIf x = 10, y = 40, and dx = 1:0 = 20 (10)^-β
(40)^β
(1) + 40 (10)^β
(40)^-β
dy0 = 20 (4)^β
+ 40 (1/4)^β
dy-20 (4)^β
= 40 (1/4)^β
dy-20 (4)^β
(1/4)^β
= 40 (1/4)^β
(1/4)^β
dy-20 = 40 (1/4) dy-20 = 10 dydy = -2The manufacturer should decrease the level of unskilled labor by 2 for the total output to stay the same.