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5.1 Quadratic functions  (Page 5/15)

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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
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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 ) .

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
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A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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8m/s²
Aishat
What is Thermodynamics
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velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
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A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
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I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
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Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
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Another formula for Acceleration
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
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What is specific heat capacity
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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