Question 1192771
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Method 1:
Use your favorite graphing tool to plot the two equations
y = -0.006x^2+32x-10,000
y = 30,000


Here's what it looks like in Desmos which is a free online graphing calculator
<a href = "https://www.desmos.com/calculator/tnzwh3datm">https://www.desmos.com/calculator/tnzwh3datm</a>
Take note of how the x and y axis scale is set up.


After plotting both the parabolic curve and straight line, click on where they intersect or cross.
Desmos will display the (x,y) coordinates of each intersection point.


The two intersection points are:
(2000, 30000)
(3333.333, 30000) approximately
When doing ordered pair notation, do NOT have a comma in the number "30000" aka "thirty thousand". This is because the comma is used to separate the x and y coordinates.


The x coordinate of the left-most point is what we're after.
This is the smaller x value such that y = 30,000.
In other words, it's the minimum amount of items to sell to achieve a revenue of $30,000.


Answer: <font color=red>2000 units</font>.


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Method 2:
Instead of graphing two equations, we can graph one equation.


Since we want the revenue R(x) to be $30,000, this means:
R(x) = 30,000
-0.006x^2+32x-10,000 = 30,000
-0.006x^2+32x-10,000 - 30,000 = 0
-0.006x^2+32x-40,000 = 0


The single equation to graph is
y = -0.006x^2+32x-40,000


When graphing this, we're looking for the x intercepts which make y = 0. Therefore, we're looking where the curve either crosses the x axis or touches it to bounce away.


Graph:
<a href = "https://www.desmos.com/calculator/vuz7masgvd">https://www.desmos.com/calculator/vuz7masgvd</a>
As the graph shows, the x coordinates of the x intercepts are the same as the results of method 1. The only difference is now that y = 0 instead of y = 30,000.


We'll arrive at the same final answer of <font color=red>2000 units</font> after selecting the smaller root.


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Method 3:
This method won't use any graphs. Instead, I'll use the quadratic formula.


R(x) = 30,000
-0.006x^2+32x-10,000 = 30,000
-0.006x^2+32x-10,000 - 30,000 = 0
-0.006x^2+32x-40,000 = 0


That final equation is in the form ax^2+bx+c = 0 where,
a = -0.006
b = 32
c = -40,000


Plugging those three items into the quadratic formula yields:
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(32)+-sqrt((32)^2-4(-0.006)(-40000)))/(2(-0.006))}}}


{{{x = (-32+-sqrt(64))/(-0.012)}}}


{{{x = (-32+-   8)/(-0.012)}}}


{{{x = (-32+8)/(-0.012)}}} or {{{x = (-32-8)/(-0.012)}}}


{{{x = (-24)/(-0.012)}}} or  {{{x = (-40)/(-0.012)}}}


{{{x = 2000}}} or  {{{x = 3333.333}}} approximately


Like before, we pick the smaller x solution to get <font color=red>2000 units</font> as the final answer.
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