The Yellow Pencil Company sells the best pencils ever made. If the cost of producing
x hundred pencils is given by C x x x ( ) 0.4 29 44 = − + 2 and if the revenue generated from the
sale of x hundred pencils is R x x x ( ) 0.6 19 = − + 2 . Money is in thousands of dollars.
(a) Show all work to find the profit function, P(x)
(b) Show all work (use the formula) to find the vertex of the profit function and
interpret its meaning in the context of this problem.
To find the profit function, P(x), we need to subtract the cost function from the revenue function.
(a) Profit function, P(x) = Revenue function - Cost function
Given:
Cost function: C(x) = 0.4x^3 - 29x^2 + 44x + 2
Revenue function: R(x) = 0.6x^3 - 19x^2 + 2
Substituting the cost and revenue functions into the profit formula:
P(x) = R(x) - C(x)
P(x) = (0.6x^3 - 19x^2 + 2) - (0.4x^3 - 29x^2 + 44x + 2)
Simplifying the equation by combining like terms:
P(x) = 0.6x^3 - 19x^2 + 2 - 0.4x^3 + 29x^2 - 44x - 2
P(x) = 0.2x^3 + 10x^2 - 44x
So, the profit function is P(x) = 0.2x^3 + 10x^2 - 44x.
(b) To find the vertex of the profit function, we need to find its maximum point. The formula for the vertex of a quadratic function (ax^2 + bx + c) in the form f(x) = ax^2 + bx + c is:
x = -b/2a
In this case, the profit function P(x) is a cubic function (0.2x^3 + 10x^2 - 44x) rather than a quadratic function, but we can still find the vertex using the same formula.
Comparing the profit function with the general quadratic function form, we have:
a = 0.2
b = 10
Using the formula x = -b/2a:
x = -(10) / (2 * 0.2)
x = -50 / 0.4
x = -125
Interpreting the meaning of the vertex:
The vertex of the profit function is at x = -125. In the context of the problem, this means that the maximum profit is achieved when x = -125, which indicates the number of hundred pencils produced/sold. However, since we cannot have a negative number of pencils produced/sold, we can interpret this as the maximum profit occurring when 125 hundred pencils are produced and sold.