SOLUTION: The profit of a company is modeled by {{{y=-0.01x^2+5x-400}}} how many items must they sell to make a positive profit?

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Question 138678: The profit of a company is modeled by y=-0.01x%5E2%2B5x-400 how many items must they sell to make a positive profit?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y=-0.01x%5E2%2B5x-400 Start with the given equation


0=-0.01x%5E2%2B5x-400 Set the equation equal to zero


x=100 or x=400 Solve for x (if you need help with this step, let me know). So the critical values are x=100 and x=400



-0.01x%5E2%2B5x-400%3E0 Set the expression greater than zero




Now set up a number line and plot the critical values on the number line

number_line%28+600%2C+80%2C+420%2C100%2C400%29



So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than 100 (notice how it's to the left of the leftmost endpoint):

So let's pick x=99

-0.01x%5E2%2B5x-400%3E0 Start with the given inequality


-0.01%2899%29%5E2%2B5%2899%29-400%3E+0 Plug in x=99


-3.01%3E+0 Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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Let's pick a test value that is in between 100 and 400:

So let's pick x=250

-0.01x%5E2%2B5x-400%3E0 Start with the given inequality


-0.01%28250%29%5E2%2B5%28250%29-400%3E+0 Plug in x=250


225%3E+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





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Let's pick a test value that is greater than 400 (notice how it's to the right of the rightmost endpoint):

So let's pick x=401

-0.01x%5E2%2B5x-400%3E0 Start with the given inequality


-0.01%28401%29%5E2%2B5%28401%29-400%3E+0 Plug in x=401


-3.01%3E+0 Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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Summary:

So the solution in interval notation is:


()



So the company needs to sell at least 100 and at most 400 widgets to get a positive profit.