SOLUTION: Please help with this question. The demand function for a new DVD is p(x)=-2.5x+17.5 where p(x) represents the selling price, in thousands of dollars, and x is the number of DVDs

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Please help with this question. The demand function for a new DVD is p(x)=-2.5x+17.5 where p(x) represents the selling price, in thousands of dollars, and x is the number of DVDs      Log On


   

Question 1137592: Please help with this question.
The demand function for a new DVD is p(x)=-2.5x+17.5 where p(x) represents the selling price, in thousands of dollars, and x is the number of DVDs sold, in thousands.
a) Determine the revenue function.
b) Determine the maximum revenue.
c) Determine the number of DVDs that need to be sold to reach the maximum revenue.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

+p%28x%29=-2.5x%2B17.5 where
p%28x%29 represents the selling price, in thousands of dollars, and
x is the number of DVDs sold, in thousands.


a) Determine the revenue function.
+Revenue+=+Price+%2A+Quantity
R%28x%29=p%28x%29%2Ax
R%28x%29=%28-2.5x%2B17.5%29x
R%28x%29=-2.5x%5E2%2B17.5x

b) Determine the maximum revenue.
R%28x%29=-2.5x%5E2%2B17.5x............complete square to find vertex, the vertex of a quadratic parabola is the highest or lowest point, the maximum or minimum
R%28x%29=-2.5%28x%5E2-7x%29
R%28x%29=-2.5%28x%5E2-7x%2Bb%5E2%29-%28-2.5%29b%5E2
R%28x%29=-2.5%28x%5E2-7x%2Bb%5E2%29%2B2.5b%5E2........b=7%2F2=3.5
R%28x%29=-2.5%28x%5E2-7x%2B3.5%5E2%29%2B2.5%2A3.5%5E2
R%28x%29=-2.5%28x-3.5%29%5E2%2B30.625
the maximum is at (3.5,30.625)
or, this way:
derivate R%28x%29:
R%2Fdx=-2.5%2A2x%2B17.5
-5x%2B17.5=0
5x=17.5
x=3.5
find r%283.5%29:
R%283.5%29=-2.5%2A3.5%5E2%2B17.5%2A3.5
R%283.5%29=-30.625%2B61.25
R%283.5%29=30.625->the maximum revenue

c) Determine the number of DVDs that need to be sold to reach the maximum revenue.

since the maximum is at vertex (3.5,30.625), the number of DVDs that need to be sold to reach the maximum revenue is x=3.5; so, in thousands, the number of DVDs that need to be sold to reach the maximum revenue is 3500