Question 1137592

{{{ 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.

{{{ Revenue = Price * Quantity}}}

{{{R(x)=p(x)*x}}}

{{{R(x)=(-2.5x+17.5)x}}}

{{{R(x)=-2.5x^2+17.5x}}}


b) Determine the maximum revenue.

{{{R(x)=-2.5x^2+17.5x}}}............complete square to find {{{vertex}}}, the vertex of a quadratic parabola is the highest or lowest point, the maximum or minimum 

{{{R(x)=-2.5(x^2-7x)}}}

{{{R(x)=-2.5(x^2-7x+b^2)-(-2.5)b^2}}}

{{{R(x)=-2.5(x^2-7x+b^2)+2.5b^2}}}........{{{b=7/2=3.5}}}

{{{R(x)=-2.5(x^2-7x+3.5^2)+2.5*3.5^2}}}

{{{R(x)=-2.5(x-3.5)^2+30.625}}}

the maximum is at ({{{3.5}}},{{{30.625}}})

or, this way:

derivate {{{R(x)}}}:

{{{R/dx=-2.5*2x+17.5}}}

{{{-5x+17.5=0}}}

{{{5x=17.5}}}

{{{x=3.5}}}

find {{{r(3.5)}}}:

{{{R(3.5)=-2.5*3.5^2+17.5*3.5}}}

{{{R(3.5)=-30.625+61.25}}}

{{{R(3.5)=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}}}