Respuesta :
Multiplying the two pieces of the function together:
[tex]R(x)=500*50 + (-5x)(500) + 125x(50) + 125(-50) \\=25000 -2500x+6250x-625x^2 \\=25000+3750x-625x^2 \\=-625x^2+3750x+25000[/tex]. This is a quadratic function, and since the coefficient of x² is negative we know it is opening downward and thus has a maximum. We can find the maximum by finding the vertex. First we find the axis of symmetry:
[tex]x=\frac{-b}{2a} \\ \\=\frac{-3750}{2*(-625)} \\ \\=\frac{-3750}{-1250}=3[/tex]. This tell us there will need to be 3 $5 decreases in price, or a $15 decrease, to maximize the function. $50-$15=$35.
We now plug our 3 in for x in our function:
[tex]R(x)=-625-(3^2)+3750(3)+25000 \\=-625(9)+11250+25000 \\=-5625+36250 \\=30,625[/tex]
If they decrease the price their total revenue will be $30,625.
Looking at the function we know that they sell an extra 125 yearbooks for every $5 decrease in price. There were 3 $5 decreases, so 3(125) = 375 extra yearbooks. Add this to the original 500 and we have 375+500=875 yearbooks.
[tex]R(x)=500*50 + (-5x)(500) + 125x(50) + 125(-50) \\=25000 -2500x+6250x-625x^2 \\=25000+3750x-625x^2 \\=-625x^2+3750x+25000[/tex]. This is a quadratic function, and since the coefficient of x² is negative we know it is opening downward and thus has a maximum. We can find the maximum by finding the vertex. First we find the axis of symmetry:
[tex]x=\frac{-b}{2a} \\ \\=\frac{-3750}{2*(-625)} \\ \\=\frac{-3750}{-1250}=3[/tex]. This tell us there will need to be 3 $5 decreases in price, or a $15 decrease, to maximize the function. $50-$15=$35.
We now plug our 3 in for x in our function:
[tex]R(x)=-625-(3^2)+3750(3)+25000 \\=-625(9)+11250+25000 \\=-5625+36250 \\=30,625[/tex]
If they decrease the price their total revenue will be $30,625.
Looking at the function we know that they sell an extra 125 yearbooks for every $5 decrease in price. There were 3 $5 decreases, so 3(125) = 375 extra yearbooks. Add this to the original 500 and we have 375+500=875 yearbooks.
