The output at a certain plant is $Q = 0.06 x ^ { 2 } + 0.15 x y + 0.05 y ^ { 2 }$ units per day,where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor used.Currently 60 hours of skilled labor and 150 hours of unskilled labor are used each day.Use calculus to estimate the change in unskilled labor that should be made to offset a 1 hour increase in skilled labor so that output will remain the same.Round your answer to two decimal places,if necessary.
A)No change
B)An increase of 1.24 hours
C)A decrease of 1.24 hours
D)It cannot be determined

Question

Quizplus · Accepted Answer

To keep the output $Q$ the same when $x$ (hours of skilled labor) increases by 1, we need to find how $y$ (hours of unskilled labor) should change. We do this by setting the total differential $dQ = 0$, which gives us the equation $0.12x dx + 0.15dx y + 0.15x dy + 0.1y dy = 0$. Substituting $x = 60$, $dx = 1$, and $y = 150$, and solving for $dy$, we find that $dy$ (the change in unskilled labor hours) is a decrease, specifically -1.24 hours.

The Output at a Certain Plant Is Q=0.06x2+0.15xy+0.05y2Q = 0.06 x ^ { 2 } + 0.15 x y + 0.05 y ^ { 2 }Q=0.06x2+0.15xy+0.05y2

The Output at a Certain Plant Is $Q = 0.06 x ^ { 2 } + 0.15 x y + 0.05 y ^ { 2 }$