Find the maximum revenue.

Question

The revenue function for a sporting goods company is given by R(x) = x ⋅ p(x) dollars where x is the number of units sold and p(x) = 700 − 0.4x is the unit price. Find the maximum revenue.

Answer 1

Hi Lucy,

 

Let's begin by composing our Revenue Function, R(x):

    R(x) = x times p(x)

    p(x)  = 700 - 0.4x

    R(x) = x(700 - 0.4x)  [Revenue Function]

 

Next, let's transform R(x) into standard quadratic form:

    R(x) = x(700 - 0.4x)

    R(x) = -04x² + 700x  [standard quadratic form]

 

Next, let's find the vertex (maximum point) using the axis of symmetry formula:

    y = ax² + bx + c  [standard quadratic form]

    x = -b/2a  [axis of symmetry formula]

    -04x² + 700x  [a = -0.4; b = 700]

    x = -700/2(-0.4)

    x = -700/-0.8

    x = 875

    y = -04x² + 700x  [x = 875]

    y = -0.4(875)² + 700(875)

    y = -0.4(765,625) + 612,500

    y = -306,250 + 612,500

    y = $306,250 (maximum revenue)

 

Below is the link to our graph of this Revenue Function:  

 

https://dl.dropbox.com/s/mw4ge5jsjgx8b2j/Graph_of_Revenue_Function.png?raw=1

 

The graph shows the vertex (maximum point) as (875,306250) meaning that a maximum revenue of $306,250 is generated when 875 units are sold.  This confirms our algebraic solution above.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.