Maximum weekly revenue of an item is the maximum price which can be generated by an owner at certain number of item at fixed unit price. By decreasing the price of the DVDs by 7 dollars the owner can generate the maximum weekly revenue of $4225.
Given-
A video store sells about 150 DVDs a week at a price of $20 each.
What is maximum weekly revenue?
Maximum weekly revenue of an item is the maximum price which can be generated by an owner at certain number of item at fixed unit price.
Let x be the number of $1 decreases in price and the R be the number of weekly revenue.
As the owner estimates that for each $1 decrease in price, about 25 more DVDs will be sold each week. Thus the revenue can be given as,
[tex]R=(150+25x)(20-1x)[/tex]
[tex]R=3000+500x-25x-150x^2[/tex]
[tex]R=3000+350x-25x^2[/tex]
Divide both side by number 25.
[tex]\dfrac{R}{25} =120+14x-x^2[/tex]
[tex]\dfrac{R}{25} =-x^2+14x+120[/tex]
The above equation is the equation of the parabola in which the coefficient of the [tex]x^2[/tex] term is negative. Hence the maximum revenue for the equation is at -b/(2a).
The standard form of the parabola is,
[tex]ax^2+bx+c=0[/tex]
Compare the obtained equation with the standard equation we get,
[tex]a=-1[/tex]
[tex]b=14[/tex]
Therefore maximum revenue is,
[tex]x=\dfrac{-b}{2a}= \dfrac{-14}{2\times(-1)}[/tex]
[tex]x= 7[/tex]
Put the value of the x in the equation of revenue,
[tex]\dfrac{R}{25} =-x^2+14x+120[/tex]
[tex]\dfrac{R}{25} =-7 ^2+14\times7+120[/tex]
[tex]\dfrac{R}{25} =-49+98+120[/tex]
[tex]R =169\times 25[/tex]
[tex]R=4225[/tex]
Thus, by decreasing the price of the DVDs by 7 dollars the owner can generate the maximum weekly revenue of $4225.
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