Question 1192771: A widget manufacturer estimates that her monthly revenue can be modeled by the function R(x)=-0.006x^2+32x-10,000. What is the minimum number of items that must be sold for the revenue to equal $30,000?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
A widget manufacturer estimates that her monthly revenue can be modeled by the function
R(x)=-0.006x^2+32x-10,000. What is the minimum number of items that must be sold
for the revenue to equal $30,000?
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To answer the question, you should solve this quadratic equation
-0.006x^2 + 32x - 10000 = 30000
and take its lesser root. Reduce the equation to the standard form quadratic equation
0.006x^2 - 32x + 40000 = 0
and apply the quadratic formula. You will get the roots
 = 2000,  = 3333.33.
Your ANSWER is the value x= 2000.
Solved.
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On solving quadratic equations using the quadratic formula, see the lessons
- Introduction into Quadratic Equations
- PROOF of quadratic formula by completing the square
in this site.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
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
https://www.desmos.com/calculator/tnzwh3datm
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: 2000 units.
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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:
https://www.desmos.com/calculator/vuz7masgvd
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 2000 units 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:
 or 
 or 
 or  approximately
Like before, we pick the smaller x solution to get 2000 units as the final answer.
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