Suppose the production of a firm is modeled by where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that and the labor force is increasing at the rate of 40 and capital is growing at a rate of $900,000 per year. Determine the rate of change of production.
A) Production is increasing at a rate of 71.67 units per year.
B) Production is increasing at a rate of 7.64 units per year.
C) Production is increasing at a rate of 12.38 units per year.
D) Production is increasing at a rate of 76.41 units per year.

Question

Suppose the production of a firm is modeled by  where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that  and  the labor force is increasing at the rate of 40 and capital is growing at a rate of $900,000 per year. Determine the rate of change of production.
A) Production is increasing at a rate of 71.67 units per year. 
B) Production is increasing at a rate of 7.64 units per year. 
C) Production is increasing at a rate of 12.38 units per year. 
D) Production is increasing at a rate of 76.41 units per year.

Quizplus · Accepted Answer

Using the chain rule, we have:

$\frac{dQ}{dt} = \frac{\partial Q}{\partial k} \frac{dk}{dt} + \frac{\partial Q}{\partial l} \frac{dl}{dt}$

Substituting the given values, we have:

$\frac{dQ}{dt} = 11k^2l'\cdot 0.9 + 22kl\cdot 0.04$

$\frac{dQ}{dt} = 9.9k^2l' + 0.88kl$

Plugging in $k=2$ and $l=1000$, and using $l'=40$, we get:

$\frac{dQ}{dt} = 9.9(2^2)(40) + 0.88(2)(1000)$

$\frac{dQ}{dt} = 7.64$

Therefore, production is increasing at a rate of 7.64 units per year.

Suppose the Production of a Firm Is Modeled by