The Yellow Pencil Company sells the best pencils ever made. If the cost of producing
x hundred pencils is given by C(x)=0.4x^2-29x+44 and if the revenue generated from the sale of x hundred pencils is R(x)=-0.6x^2+19x. 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.
P(x) = R(x) - C(x)
= .4x^2 - 29x + 44 +.6x^2 - 19x
= x^2 - 48x + 44
this parabola cuts the x-axis at
x = 47.065 and a negative
Since x represents pencils, x>0
So this parabola will be above the x-axis for x>47
x of vertex is -b/(2a)
= 48/2 = 24
P(24) = -532
the vertex is (24,-532)
they will not make any profit until x > 47
(a) To find the profit function, we need to subtract the cost function from the revenue function.
Profit = Revenue - Cost
So, we need to subtract C(x) from R(x).
P(x) = R(x) - C(x)
P(x) = (-0.6x^2 + 19x) - (0.4x^2 - 29x + 44)
P(x) = -0.6x^2 + 19x - 0.4x^2 + 29x - 44
P(x) = -1x^2 + 48x - 44
Therefore, the profit function is P(x) = -x^2 + 48x - 44.
(b) To find the vertex of the profit function, we can use the formula x = -b/2a, where the profit function is in the form ax^2 + bx + c.
In this case, a = -1 and b = 48.
x = -b/2a
x = -(48) / (2 * (-1))
x = -48 / -2
x = 24
Now we need to find the corresponding y-value by substituting x = 24 into the profit function.
P(24) = -(24)^2 + 48(24) - 44
P(24) = -576 + 1152 - 44
P(24) = 532
The vertex of the profit function is (24, 532).
In this context, the vertex represents the level of production (x) that maximizes the profit (P). So, when the company produces 24 hundred pencils, it maximizes its profit, which is equal to $532,000.