A marketing department estimates that demand for a product is given by P=100-0.0001x, where P is price per unit and X is number of units. The cost C of producing x units is given by
C=350,000+25x
And the profit P for producing x units is given by
P= R-C=xp=C.
Sketch the graph of the profit function.
Use a graphing utility to estimate the number of units that would produce a maximum profit ( to nearest thousand).
So I got as far as p=x(100-0.0001x)-350,000+25x.
When I graph that, it doesn't match what the options of graphs on my homework are.
no
To sketch the graph of the profit function, you need to simplify the equation you derived:
P = x(100 - 0.0001x) - 350,000 + 25x
First, distribute x into the parentheses:
P = 100x - 0.0001x^2 - 350,000 + 25x
Combine like terms:
P = -0.0001x^2 + 125x - 350,000
Now, you have a quadratic equation in a standard form: P = ax^2 + bx + c. Comparing it to the equation, you can identify the coefficients:
a = -0.0001
b = 125
c = -350,000
To graph the function, use a graphing utility or software.
Next, to estimate the number of units that would produce maximum profit, you can either analyze the graph or use calculus. Here, we will use calculus to find the x-coordinate of the vertex, which represents the maximum profit.
The x-coordinate of the vertex of a quadratic function is given by:
x = -b / (2a)
In our case:
x = -(125) / (2(-0.0001)) ≈ 625,000
So, producing around 625,000 units of the product would likely result in the maximum profit.
Note: Make sure to double-check your calculations and verify your answer by rechecking the graph or using alternative methods if possible.