Answer:
Part A)
Part B)
The daily operating cost decreases by about $143 per extra worker.
Step-by-step explanation:
We are given the equation:
Where <em>P</em> is the number of eggs laid, <em>x</em> is the number of workers, and <em>y</em> is the daily operating budget (assuming in US dollars $).
A)
We want to find dy/dx.
So, let’s find our equation in terms of <em>x</em>. We can raise both sides to 10/7. Hence:
Simplify:
Divide both sides by<em> </em>the <em>x</em> term to acquire:
Take the derivative of both sides with respect to <em>x: </em>
Apply power rule. Note that P is simply a constant. Hence:
Simplify. Hence, our derivative is:
Part B)
We want to evaluate the derivative when <em>x</em> is 30 and when <em>y</em> is $10,000.
First, we will need to find <em>P</em>. Our original equations tells us that:
Hence, at <em>x</em> = 30 and at <em>y</em> = 10,000, <em>P </em>is:
Therefore, for our derivative, we will have:
Use a calculator. So:
Our derivative is given by dy/dx. So, it represents the change in the daily operating cost over the change in the number of workers.
So, when there are 30 workers with a daily operating cost of $10,000 producing a total of about 1750 eggs, the daily operating cost decreases by about $143 per extra worker.