Paramount Electronics has an annual profit given by P=-150,000+6,000q -0.1q dollars Where q is the number of laptop computers it sells each year. The number of laptop computers it can make and sell each year depends on the number n of electrical engineers paramount employees, according to the equation q 35n+0.02n’. Find dP at n12 and interpret the result
The Correct Answer and Explanation is :
To solve this, we first need to express the profit ( P ) in terms of the number ( n ) of electrical engineers. From the information provided:
[ q = 35n + 0.02n^2 ]
Substituting ( q ) into the profit function ( P ):
[ P = -150,000 + 6,000(35n + 0.02n^2) – 0.1(35n + 0.02n^2)^2 ]
We need to find the rate of change of ( P ) with respect to ( n ), denoted as ( \frac{dP}{dn} ), and then evaluate it at ( n = 12 ).
First, expand the expression for ( P ):
[ P = -150,000 + 6,000(35n + 0.02n^2) – 0.1(1225n^2 + 1.4n^3 + 0.0004n^4) ]
Expanding and simplifying:
[ P = -150,000 + 210,000n + 120n^2 – 122.5n^2 – 0.14n^3 – 0.00004n^4 ]
[ P = -150,000 + 210,000n – 2.5n^2 – 0.14n^3 – 0.00004n^4 ]
Now, we differentiate ( P ) with respect to ( n ):
[ \frac{dP}{dn} = 210,000 – 5n – 0.42n^2 – 0.00016n^3 ]
Plugging ( n = 12 ) into this derivative:
[ \frac{dP}{dn} = 210,000 – 5(12) – 0.42(12)^2 – 0.00016(12)^3 ]
Let’s calculate this value using Python:
n = 12
dP_dn = 210000 - 5*n - 0.42*n**2 - 0.00016*n**3
dP_dnAfter obtaining the result, we can interpret it.
Interpretation of Result:
The derivative ( \frac{dP}{dn} ) at ( n = 12 ) tells us the rate at which the profit ( P ) changes with respect to the number of engineers ( n ). If ( \frac{dP}{dn} ) is positive, it implies that hiring more engineers (around 12 engineers) will increase the profit. If negative, it means that hiring more engineers will decrease the profit. The exact value provides the rate of change of profit per additional engineer hired at ( n = 12 ).
Now, let’s calculate the derivative’s value and then create an image that represents this interpretation visually.
The derivative of the profit function ( \frac{dP}{dn} ) at ( n = 12 ) is approximately 209,879.24. This means that at the point where Paramount Electronics employs 12 engineers, the profit increases by about $209,879 per additional engineer hired.
Interpretation:
The positive value of the derivative indicates that hiring more electrical engineers beyond 12 is currently beneficial for increasing Paramount Electronics’ profit. The large magnitude of this rate of change suggests that each additional engineer significantly contributes to the profitability, making it a strategic decision to possibly expand the engineering workforce.
This result highlights the importance of workforce sizing in production capacities and the direct impact of skilled labor on financial outcomes. Decisions on hiring should consider this high sensitivity of profit to the number of engineers, especially around this level of workforce size. Paramount Electronics might look into whether this trend continues beyond a certain number of engineers or if there is an optimal number where profit no longer increases with additional hires at such a substantial rate.
Now, let’s create a visual representation of this concept:
The image will depict a graph showing the profit curve as it changes with the number of engineers, highlighting the point at ( n = 12 ) with an upward arrow and a text annotation of the derivative value to emphasize the profit increase per additional engineer.
Let’s generate that image.
Here is the visual representation of the profit function for Paramount Electronics, showing how the profit increases with the number of engineers, particularly highlighting the situation at ( n = 12 ). This graph illustrates the significant positive impact of hiring additional engineers on the company’s profitability.
