Question 389610: The revenue from selling x units of a product is given by
y = -0.0002x^2 + 20x. How many units must be sold in
order to have the greatest revenue? (Find the x-coordinate
of the vertex of the parabola.)
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! y=-0.0002x^(2)+20x
To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x. In this problem, add (-50000)^(2) to both sides of the equation.
y=-0.0002(x^(2)-100000x+2500000000)-(-0.0002)(0+2500000000)
Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
y=-0.0002(x^(2)-100000x+2500000000)-(-0.0002)(2500000000)
Factor the perfect trinomial square into (x-50000)^(2).
y=-0.0002((x-50000)^(2))-(-0.0002)(2500000000)
Factor the perfect trinomial square into (x-50000)^(2).
y=-0.0002(x-50000)^(2)-(-0.0002)(2500000000)
Multiply -0.0002 by 2500000000 to get -500000.
y=-0.0002(x-50000)^(2)-(-500000)
Multiply -1 by each term inside the parentheses.
y=-0.0002(x-50000)^(2)+500000
This is the form of a paraboloa. Use this form to determine the values used to find vertex and x-y intercepts.
y=a(x-h)^(2)+k
Use the standard form to determine the vertex and x-y intercepts.
a=-0.0002_k=500000_h=50000
The vertex of a parabola is (h,k).
Vertex: (50000,500000)
This formula is used to find the distance from the vertex to the focus.
(1)/(4p)=a
Substitute the value of a into the formula.
(1)/(4p)=-0.0002
Solve the equation for p.
p=-1250
Add p to the vertex to find the focus. If the parabola points up or down add p to the y-coordinate of the vertex, if it points left or right add it to the x-coordinate.
Focus=(50000,500000-1250)
Find the focus.
Focus=(50000,498750)
A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix).
y=500000-(-1250)
Find the directrix.
Directrix: y=501250
The axis of symmetry is the line that passes through the vertex and focus. The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.
Axis of Symmetry: x=50000
These values represent the important values for graphing and analyzing a parabola.
Vertex: (50000,500000)_Focus: (50000,498750)_Directrix: y=501250_Axis of Symmetry: x=50000
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