Suppose the output at a certain factory is $Q = 4 x ^ { 2 } + 3 x ^ { 3 } y ^ { 3 } + y ^ { 5 }$ units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 30 hours of skilled labor and 10 hours of unskilled labor. Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level. Round you answer to two decimal places, if necessary.
A) -0.33 hours
B) -1 hours
C) -3.01 hours
D) 3.01 hours

Question

Quizplus · Accepted Answer

To maintain the output level when skilled labor x increases by 1 hour, we need to find how unskilled labor y should change. We use the total differential of the output function $Q = 4x^2 + 3x^3y^3 + y^5$, which is $dQ = (8x + 9x^2y^3)dx + (9x^3y^2 + 5y^4)dy$. Setting $dQ = 0$ because the output is to remain constant, and substituting $x = 30$ and $y = 10$, we solve for $dy/dx$ to find the change in y for a unit change in x. The calculation yields $dy/dx \approx -0.33$, indicating that for every 1-hour increase in skilled labor, unskilled labor should decrease by approximately 0.33 hours to maintain the same output level.

Suppose the Output at a Certain Factory Is Q=4x2+3x3y3+y5Q = 4 x ^ { 2 } + 3 x ^ { 3 } y ^ { 3 } + y ^ { 5 }Q=4x2+3x3y3+y5

Suppose the Output at a Certain Factory Is $Q = 4 x ^ { 2 } + 3 x ^ { 3 } y ^ { 3 } + y ^ { 5 }$