Expert: Ahmed Salami - 10/8/2011

Question
The output at a certain factory is Q (L)=600L^2/3 units, where L is the size of the labor force. The manufacturer wishes to increase output by 1%. Use calculus to estimate the percentage increase in labor that will be required.

The answer is 1.5% but I need to know the steps to get there.  Thanks!

Answer
Hi Kristen,
When changes in variables are small we can use the linear approximation formula;
Δy ≈ dy/dx . Δx
For the function Q = 600L^(2/3)
dQ/dL = 600 . (2/3)L^(-1/3) = 400/L^(1/3)
and so
ΔQ ≈ dQ/dL . ΔL = 400/L^(1/3) ΔL
To write in fractional (or percentage) form, we divide both sides by Q and have
ΔQ/Q ≈ 400/L^(1/3) ΔL / Q = 400/L^(1/3) ΔL / 600L^(2/3) = (2/3).(ΔL/L)
ΔL/L = (3/2) ΔQ/Q
and we can see that the fractional change in L (i.e ΔL/L) is 1.5 times the fractional change in Q.
Therefore,
ΔL/L = (3/2) . 1%
= 1.5%

Regards