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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

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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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
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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.
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Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
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