Bulldozers

Question

The expresion P(x) = 2500x - 2x² describes the profit of a company that customizes bulldozers when it customizes x bulldozers in a month. How many bulldozers per month must the company customize to make the maximum possible profit? What is the maximum profit? Describe a reasonable domain and range for this problem.

Answer 1

Hi Haley,

 

Let's begin by writing the Profit Function, P(x), in standard quadratic form (ax² + bx + c):

    P(x) = -2x² + 2500x (a = -2; b = 2500; c = 0)

 

Now let's use the axis of symmetry formula (x = -b/2a) for a parabola

to calculate the x coordinate of the vertex (maximum point) to determine the number of bulldozers (x) needed to be customized for maximum profit:

    x = -b/2a

    x = -2500/[2(-2)]

    x = 2500/4

    x = 625 customized bulldozers for maximum profit

 

Next, let's plug in 625 for x into the Profit Function, P(x), to determine the maximum profit:

    P(x) = -2x² + 2500x

    P(x) = -2(625)² + 2500(625)

    P(x) = -2(390,625) + 1,562,500

    P(x) = -781,250 + 1,562,500

    P(x) = $781,250 (maximum profit)

 

Finally, let's set the Profit Function, P(x), to zero and find the zeroes (roots) of the function in order to determine the reasonable domain and range for the Profit Function, P(x):

     P(x) = -2x² + 2500x

    -2x² + 2500x = 0

    -2x(x -1250) = 0

    -2x = 0  |  x - 1250 = 0

       x = 0  |  x = 1250

Since the goal of any company is to earn a profit, the following values are the reasonable domain and range for the Profit Function, P(x), based upon the previously determined zeroes (roots) and maximum value of the Profit Function, P(x):

    Domain:  0 < x < 1250

    Range:  0 < P(x) < 781,250

 

Below is the link to our graph of the Profit Function, P(x):

https://dl.dropbox.com/s/lnvuknqhdgyk4xj/Graph_of_Bulldozer_Profit_Function.png?raw=1

 

The graph shows the zeroes (roots) and the vertex (maximum value) of the Profit Function, P(x).  These points marked in red confirm our algebraic solution above.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan (Romans 5:8)