The annual profit, p(x), in dollars of a small company varies with the number of employees, x, as p(x) = -40x^2 + 4400x. What is the range of the number of employees for which the company’s annual profit will be at least $112,000?
p(x) = -40x^2 + 4400x
-40x^2 + 4400x >= 112000
-40x^2 + 4400x - 112000 >= 0
-40x^2 + 4400x - 112000 >= 0
divide everything by 40
-x^2 + 110x - 2800 >= 0
now times by -1 but don't forget that when you multiply an inequality by a neg, you have to turn the sign around
x^2 - 110x + 2800 <= 0
now you have to factorise to get the roots. I notice straight away that -4*-7=28 and -4+-7=-11 so I know what the factors are but if you don't see this you can use the quadratic equation.
(x-40)(x-70) <=0
When x=40, or x=70 this will equal 0
consider the graph
y=(x-40)(x-70)
the equation we want will be true if y is positive
this is the equation of a concave up parabola with roots at 40 and 70 (sketch it)
So y will be positive or zero 40<=x<=70
so to make a profit of at least $112000 the company need to have between 40 and 70 employees.
If anything doesn't make sense, please post again.
+118653 
