Question

The profit of a cell-phone manufacturer is found by the function y= -2x2 + 108x + 75 , where x is the cost of the cell phone. At what price should the manufacturer sell the phone tomaximize its profits? What will the maximum profit be?

Answer 1

Hello!

First, let's rewrite the function:

`y=-2x^2+108x+75`

Now, let's find each coefficient of it:

• a = -2

,

• b = 108

,

• c = 75

As we have a < 0, the concavity of the parabola will face downwards.

So, it will have a maximum point.

To find this maximum point, we must obtain the coordinates of the vertex, using the formulas below:

`\begin{gathered} X_V=-(b)/(2\cdot a) \\ \\ Y_V=-(\Delta)/(4\cdot a) \end{gathered}`

First, let's calculate the coordinate X by replacing the values of the coefficients:

`\begin{gathered} X_V=-(b)/(2\cdot a) \\ \\ X_V=-(108)/(2\cdot(-2))=-(108)/(-4)=(108)/(4)=(54)/(2)=27 \end{gathered}`

So, the coordinate x = 27.

Now, let's find the y coordinate:

`\begin{gathered} Y_V=-(\Delta)/(4\cdot a) \\ \\ Y_V=-(b^2-4\cdot a\cdot c)/(4\cdot a) \\ \\ Y_V=-(108^2-4\cdot(-2)\cdot75)/(4\cdot(-2)) \\ \\ Y_V=-(11664+600)/(-8)=(12264)/(8)=1533 \end{gathered}`

The coordinate y = 1533.

Answer:

The maximum profit will be 1533 (value of y) when x = 27.