A company plans to sell pens for $2 each. The company's financial planner estimates that the cost, y, of manufacturing the pens is a quadratic function with a y-intercept of 120 and a vertex of (250, 370). What is the minimum number of pens the company must sell to make a profit?

173

174

442

443

Good evening (here it is 23h45),

1) we must find the cost equation (x is the number of pens)

y=ax²+bx+c

1) if x=0, y=120 = > c=120

2) the vertex (250,370)

y=ax²+bx+120

y'=2ax+b

y'=0 = = >2ax+b=0==> x=-b / (2a) = 250

==>500 a=-b (1)

if x=250, y=a*250²+b*250+120=370

==>250a+b=1 (2)

We put the value of b in (2)

==>250a-500a=1==>a=-1/250

So b=-500a=-500 * (-1/250) = 2

Profit = 0 = owning - cost (sorry for the words i don't know)

B (x) = 2*x - (-x²/250+2x+120) = 0

=>x²/250-120=0

x²=30000

x=173.20508 ...

==> 174

The equation of the cost is : y=-x/500+2x+120