Question 1196129
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The tutor @ikleyn has the right idea, but made an error when going from
2040x - 1500x - 3,000,000 = 0.02x^2 + 40x
to
x^2 + 500x - 3,000,000 = 0


This is how I would continue the steps to solve for x
2040x - 1500x - 3,000,000 = 0.02x^2 + 40x
2040x - 1500x - 3,000,000 - 0.02x^2 - 40x = 0
-0.02x^2 + 500x - 3,000,000 = 0


Now apply the quadratic formula with
a = -0.02
b = 500
c = -3,000,000
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-500+-sqrt((500)^2-4(-0.02)(-3000000)))/(2(-0.02))}}}


{{{x = (-500+-sqrt(10000))/(-0.04)}}}


{{{x = (-500+-  100)/(-0.04)}}}


{{{x = (-500+100)/(-0.04)}}} or {{{x = (-500-100)/(-0.04)}}}


{{{x = (-400)/(-0.04)}}} or  {{{x = (-600)/(-0.04)}}}


{{{x = 10000}}} or  {{{x = 15000}}}


The two potential answers are 
x = 10,000
x = 15,000


However we need to check them both


Let's check x = 10,000 first
If 10,000 units were sold last year, at $3 each, then the company earns 3*10,000 = 30,000 dollars in sales
Divide the net income for last year (4500) over the sales (30,000) and we'll get the profit margin for last year
margin = (4500)/(30,000) = 0.15
The profit margin for last year is 0.15, ie 15% of sales is net income
However, we've hit the ceiling since it states "The company never has had a margin of profit greater than 0.15"
So there's no way to have a profit margin increase of 0.02 to get to 0.15+0.02 = 0.17 for this current year.
Therefore, the value x = 10,000 is not a solution. 



Now check x = 15,000
last year's sales = 15,000*3 = 45,000
last year profit margin = (net income)/(sales) = 4500/(45,000) = 0.10
The profit margin for last year was 0.10
So far so good since we're not at the ceiling of 0.15
this year's sales = (15,000+2,000)*(3+0.50) = 17,000*3.50 = 59,500
this years profit margin = (7140)/(59500) = 0.12 which is 0.02 more than 0.10
This confirms that x = 15,000 works as a solution



Answers: 
15,000 units sold last year
17,000 units sold this year
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