*Determining maximum or minimum related to operations with numbers.

Two number have a difference of 18. Does the sum of their square have a maximum or a minimum value? Determine this value and the two numbers.

Solve:
Write an equation for the sum of the squares, S.
Let one number be represented by x.
The difference between the two number is 18.
So, the other number is x+18.
The squares of the numbers are and
An equation is:



The coefficient of is positive, so the graph has a minimum point and a minimum value. Complete the square to determine the coordinates of the vertex.




The sum of the squares is a minimum.
From the equation, the coordinates of the vertex are: (-9, 162)
So, the minimum value is 162.
The x-coordinate of the vertex, which is -9, is one number.
The other number is: -9 + 18 = 9

*Modelling a problem for maximizing profit.

Select Audio Company sells an MP3 player for $75. At that price, the company sells approximately 1000 players per week. The company predicts that for every $5 increase in price, it will sell 50 fewer MP3 players.
a) Which price for an MP3 player will maximize the revenue?
b) What is the maximum revenue?

Solve:
a) Let x represent the number of $5 increases in the price of an MP3 player.
When the price increases by $5 x times:
The price of an MP3 player is 75 + 5x
The number of MP3 players sold is 1000 – 50x
The revenue is: (75 + 5x)(1000 – 50x)
Let the revenue be R dollars.
An equation is: R= (75 + 5x)(1000 – 50x)
From this factored form of the equation, we can calculate the x-intercepts by finding the roots when R=0, which are: -15 and 20.
The axis of symmetry is a vertical line halfway between x = -15 and x = 20
The constant term in the equation is the mean of the x-intercepts:
So, the equation of the axis of symmetry, the number of $5 increase that maximizes the revenue, is: x=2.5
The number of increase is a whole number, so round 2.5 to 2 or 3 (since the distance from 2 to 2.5 and from 3 to 2.5 is the same: 0.5 units)
Two increases of $5 in the cost of an MP3 player mean that the price will now cost: 75 + 5(2)= $85
similarly, Three increases of $5 in the cost mean that the price will now cost: 75 + 5(3) = $90

b) Substitute either x=2 or x=3 in R= (75 + 5x)(1000 – 50x) to determin the corresponding revenue.
R= (75 + 5(2))(1000 – 50(2))= $76500
or
R= (75 + 5(3))(1000 – 50(3))= $76500