|
Question 563998: Hi, I need help trying to understand how to do this problem. I have seen this problem on this site, but I still can't understand how to do it. Can somebody please explain this to me. Your company would like to know how sales levels affect profits. If too few items are sold, then there is a loss. Even if too many items are sold, however, the company can lose money (likely because of low pricing). It is good to know how many items can be sold to make a profit.
P(x)= -x^2 +110x - 1000
This function can be used to compute the profit (in thousands of dollars)from producing and selling a certain number, x, of thousands of smartphones.
A. Compute the following: P(5), P(50), P(120). Then interpret the results.
B. Then how do I graph the function? C. Discuss and interpret the meaning where the profit function crosses the x-axis. Also, discuss where the graph is above and below the x-axis, explaining what that means in terms of profitability.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A. It's not hard to do the computations requested
P(5)=  = -475
P(50)=  = 2000
P(120)=  = -2200
To start manufacturing a new item, you need to design it, buy and install the machinery needed to make it, train people for the production work, and buy a starter set of supplies. There are also fixed cost like rent, and utilities. If you only sell very few items you cannot cover your initial and fixed costs. You are in the situation illustrated by P(5)= -475, you are loosing money (P<0).
If you sell enough to cover starting costs and ongoing costs (like extra supplies to produce more items), you may start showing an actual profit (P>0), and be in the situation illustrated by P(50)= 2200.
If you get overly optimistic and produce too much of the item you will "saturate the market," running out of customers willing to pay the set price. You would end up having to lower the price, maybe even below your costs, or you may be unable to sell all of your product, or both. In either case, you'll be operating at a loss (P<0), as is the case for P(120)=-2200.
B.The function  is a quadratic function.
Quadratic functions can be written as
 .
Their graph, called a parabola has an axis of symmetry line with equation  .
At that value of  is the vertex of the parabola.
It is a maximum of the function (if  ), or a minimum (if  ).
For graphing such a function, we could plot the vertex, a few points to one side, and the symmetrical points to the other side.
In this case,  and the function has a maximum.
The axis of symmetry and x-coordinate of the vertex/maximum is
P(55)=  = 2025 gives us the maximum/vertex at point (55,2025).
Other points:
P(10)=  = 0 gives us point (10,0), and the symmetrical point (100,0).
P(20)=  = 800 gives us point (20,800), and the symmetrical point (90,800).
We also have points from the calculations in part A.
P(5)= -475 for point (5,-475) and the symmetrical point (105,-475)
P(50)= 2000 for point (50,2000) and the symmetrical point (60,2000)
All those points, plotted and connected give you a graph that looks like this
C. The function crosses the axis at  and 
At those two points the profit is zero. In between, the graph is above the y-axis and the business is profitable. Producing less than 10 units results in too little income for sales, and financial loss. Producing more than 100 units, results in loss too, because there are not enough buyers, and the money spent on producing so many units will not be recovered.
|
|
|
| |
