SOLUTION: 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. (

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 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. (      Log On


   

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 recieved by the manufacturer in a week.
My work- would it be 24-1 I really don't understand a & b
b. What price will maxamize the total revenue?
Please explain
I would really appreciate any help
Thank you
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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 = %28-1400%29%2F%282%2A-100%29
x = %28-1400%29%2F%28-200%29
x = +7; a $7 reduction
:
Price = 24 - 7 = $17 for max revenue
:
:
If you plotted this it would look like this:
+graph%28+300%2C+200%2C+-10%2C+25%2C+-5000%2C+30000%2C+-100x%5E2%2B1400x%2B24000%29+
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.