<< Chapter < Page Chapter >> Page >

Finding maximum revenue

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, p for price per subscription and Q for quantity, giving us the equation Revenue = p Q .

Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently p = 30 and Q = 84,000. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, p = 32 and Q = 79,000. From this we can find a linear equation relating the two quantities. The slope will be

m = 79,000 84,000 32 30 = −5,000 2 = −2,500

This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y -intercept.

Q = −2500 p + b Substitute in the point Q = 84,000  and  p = 30 84,000 = −2500 ( 30 ) + b Solve for b b = 159,000

This gives us the linear equation Q = −2,500 p + 159,000 relating cost and subscribers. We now return to our revenue equation.

Revenue = p Q Revenue = p ( −2,500 p + 159,000 ) Revenue = −2,500 p 2 + 159,000 p

We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.

h = 159,000 2 ( −2,500 ) = 31.8

The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.

maximum revenue = −2,500 ( 31.8 ) 2 + 159,000 ( 31.8 ) = 2,528,100

Finding the x - and y -intercepts of a quadratic function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y - intercept of a quadratic by evaluating the function at an input of zero, and we find the x - intercepts at locations where the output is zero. Notice in [link] that the number of x - intercepts can vary depending upon the location of the graph.

Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one –intercept, and the third parabola is of two x-intercepts.
Number of x -intercepts of a parabola

Given a quadratic function f ( x ) , find the y - and x -intercepts.

  1. Evaluate f ( 0 ) to find the y -intercept.
  2. Solve the quadratic equation f ( x ) = 0 to find the x -intercepts.

Finding the y - and x -intercepts of a parabola

Find the y - and x -intercepts of the quadratic f ( x ) = 3 x 2 + 5 x 2.

We find the y -intercept by evaluating f ( 0 ) .

f ( 0 ) = 3 ( 0 ) 2 + 5 ( 0 ) 2 = −2

So the y -intercept is at ( 0 , −2 ) .

For the x -intercepts, we find all solutions of f ( x ) = 0.

0 = 3 x 2 + 5 x 2

In this case, the quadratic can be factored easily, providing the simplest method for solution.

0 = ( 3 x 1 ) ( x + 2 )
h = b 2 a k = f ( −1 ) = 4 2 ( 2 ) = 2 ( −1 ) 2 + 4 ( −1 ) 4 = −1 = −6

So the x -intercepts are at ( 1 3 , 0 ) and ( 2 , 0 ) .

Questions & Answers

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Selected topics in algebra' conversation and receive update notifications?

Ask