SOLUTION: If the selling price x is related to profit p by the equation p=5000x−125x^2 i)For what range of values of x is the profit increasing? ii)For what range of values of x is t

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Question 973025: If the selling price x is related to profit p by the equation p=5000x−125x^2
i)For what range of values of x is the profit increasing?
ii)For what range of values of x is the profit decreasing?
iii)Determine the value of x that would yield maximum profit.
iv)Determine the maximum profit.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
If the selling price x is related to profit p by the equation p=5000x−125x^2

We can tell everything we need from the graph.  So let's graph it:

p = 5000x − 125x²

Put it in descending order of powers of x:

p = −125x² + 5000x

We recognize this as a quadratic equation whose graph is a
parabola.  Since the leading coefficient is negative it opens
downward.

The maximum point is the vertex.

We compare 

p = −125x² + 5000x

to

y = ax² + bx + c

and determine that a = -125, b = 5000 and c = 0

The formula for the x-cordinate of the vertex is -b%2F%282a%29%22%22=%22%22%28-5000%29%2F%282%2A%28-125%29%29=%28-5000%29%2F%28-250%29=20

Substituting in p = -125x² + 5000x
                p = -125(20)² + 5000(20)
                p = -125(400) + 100000
                p = -50000 + 100000
                p = 50000

So the vertex is the point (20,50000)

We determine the x and p intercepts:

To determine the x-intercepts we set p = 0

               0 = -125x² + 5000x
   125x² - 5000x = 0

Divide through by 125

        x² - 40x = 0

Factor out x

       x(x - 40) = 0

Set both the factor x and the factor x - 40 equal to 0

    x = 0;  x - 40 = 0
                 x = 40

So the x-intercepts are (0,0) and (40,0)

To find the p-intercept, we set x = 0

       p = -125(0)² + 5000(0)
       p = 0

The p-intercept is (0,0), the origin.
 


i)For what range of values of x is the profit increasing?

on the interval 0 < x < 20 

ii)For what range of values of x is the profit decreasing?

on the interval 20 < x < 40


iii)Determine the value of x that would yield maximum profit.

x = 20

iv)Determine the maximum profit.

p = 50000

Edwin