Answer:
Part A)
[tex]\displaystyle \frac{dy}{dx}=-\frac{3}{7}P^\frac{10}{7}x^{-\frac{10}{7}}[/tex]
Part B)
The daily operating cost decreases by about $143 per extra worker.
Step-by-step explanation:
We are given the equation:
[tex]\displaystyle P=x^{\frac{3}{10}}y^{\frac{7}{10}}[/tex]
Where P is the number of eggs laid, x is the number of workers, and y 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 x. We can raise both sides to 10/7. Hence:
[tex]\displaystyle P^\frac{10}{7}=\Big(x^\frac{3}{10}y^\frac{7}{10}\Big)^\frac{10}{7}[/tex]
Simplify:
[tex]\displaystyle P^\frac{10}{7}=x^\frac{3}{7}y[/tex]
Divide both sides by the x term to acquire:
[tex]\displaystyle y=P^\frac{10}{7}x^{-\frac{3}{7}}[/tex]
Take the derivative of both sides with respect to x:
[tex]\displaystyle \frac{dy}{dx}=\frac{d}{dx}\Big[P^\frac{10}{7}x^{-\frac{3}{7}}\Big][/tex]
Apply power rule. Note that P is simply a constant. Hence:
[tex]\displaystyle \frac{dy}{dx}=P^\frac{10}{7}(-\frac{3}{7})(x^{-\frac{10}{7}})[/tex]
Simplify. Hence, our derivative is:
[tex]\displaystyle \frac{dy}{dx}=-\frac{3}{7}P^\frac{10}{7}x^{-\frac{10}{7}}[/tex]
Part B)
We want to evaluate the derivative when x is 30 and when y is $10,000.
First, we will need to find P. Our original equations tells us that:
[tex]P=x^{0.3}y^{0.7}[/tex]
Hence, at x = 30 and at y = 10,000, P is:
[tex]P=(30)^{0.3}(10000)^{0.7}[/tex]
Therefore, for our derivative, we will have:
[tex]\displaystyle \frac{dy}{dx}=-\frac{3}{7}\Big(30^{0.3}(10000^{0.7})\Big)^\frac{10}{7}\Big(30^{-\frac{10}{7}}\Big)[/tex]
Use a calculator. So:
[tex]\displaystyle \frac{dy}{dx}=-\frac{1000}{7}=-142.857142...\approx-143[/tex]
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.
