Question 172694
A manufacturer charges $24 for stereo headphones and has been selling about 1000 a week. 
He estimates that for every $1 price reduction, 100 more headphones can be sold per week. 
(For example, he could sell 1100 headphones at $23 each and 1200 headphones at $22 each.)
:
a. Let 24-x be the reduced price per set of headphones. write a quadratic function that gives the total revenue received by the manufacturer in a week.
"
:
x = no. of dollar reductions in price
and
x = additional no. of 100's of units sold
:
Price = (24-x)
Units sold = 1000+100x
:
Revenue = price * units sold
:
R = (24-x) (1000+100x)
FOIL
R = 24000 + 2400x - 1000x - 100x^2
:
R = -100x^2 + 1400x + 24000; the quadratic equation for revenue
:
:
b. What price will maximize the total revenue? 
:
Since this is a quadratic equation, the axis of symmetry will give a value
for x that will make (24-x) be the price for maximum revenue
:
Find the axis of symmetry using the formula x = -b/(2a)
In this equation a=-100; b=1400
x = {{{(-1400)/(2*-100)}}}
x = {{{(-1400)/(-200)}}}
x = +7; a $7 reduction 
:
Price = 24 - 7 = $17 for max revenue
:
:
If you plotted this it would look like this:
{{{ graph( 300, 200, -10, 25, -5000, 30000, -100x^2+1400x+24000) }}}
Where x = dollar reduction from $24, and y is the total revenue
You can see max revenue occurs when x = 7 which is a unit price of $17
:
If you want to know exactly what the max revenue is, substitute 7 for x in the
original equation.