For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees (in thousands) can be approximated by

(a) Would be expected to be positive, negative or zero? Why?
(b) Find . What are the units?
© Find when and , and interpret the answer.

Let’s analyze the relationship given by the equation .

(a) To determine whether is expected to be positive, negative, or zero, we can examine the equation. As revenue increases, one would typically expect that the number of employees would also increase in order to support the increased revenue. Therefore, we can expect to be positive, indicating a direct relationship between revenue and the number of employees.

(b) To find , we first differentiate both sides of the equation with respect to . Using implicit differentiation, we have:

This gives us:

Now, solving for :

The units of are in billions of dollars (billion $) and is in thousands of employees (thousand employees), thus:

So the units for will be thousands of employees per billion dollars, or .

© Now we need to find when and . First, we can plug those values into the formula we derived:

Calculating the numerator:

Now calculating the denominator:

Now, substituting these values in,

Thus, thousands of employees per billion dollars.

Interpretation: This means that for every additional billion dollars in revenue, the company is expected to increase its workforce by approximately 3,845 employees. This relationship illustrates how crucial scaling the workforce is to meet the demands accompanied by revenue growth!