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

Answer:Option B is correctThe minimum number of pens the company must sell to make a profit is, 174.Explanation:Let x be the number of pens  and y be the cost of the pens.To find the cost of the equation.It is given that cost , y , of manufacturing the pens is a quadratic function i.,e[tex]y=ax^2+bx+c[/tex]                 ......[1]and  y-intercept of 120 which means that for x=0 , y=120 and Vertex = (250 , 370).Put x = 0  and y =120  in [1]120 = 0+0+c⇒  c= 120.Since, a quadratic function has axis of symmetry.The axis of symmetry is given by:[tex]x =\frac{-b}{2a}[/tex]                 ......[2]Substitute the value of x = 250 in [2];[tex]250 = \frac{-b}{2a}[/tex] or[tex]500a = -b[/tex]                         ......[3]Substitute the value of x=250, y =370, c =120 and b = -500 a in [1];[tex]370=a(250)^2+(-500a)(250)+120[/tex]  or[tex]250 = a(250)^2-(500a)(250)[/tex] or[tex]1 = 250a -500 a[/tex] or1 = -250 a⇒[tex]a= \frac{-1}{250}[/tex]We put the value of a in [3]So,b =-500 a= [tex]-500 \cdot \frac{-1}{250}[/tex]Simplify:b =2Therefore, the cost  price of the pens is:  [tex]y = (\frac{-1}{250})x^2+2x+120[/tex]And the selling  of the pens is 2x [ as company sell pens $ 2 each]To find the minimum number of pens the company must sell to make a profit:profit = selling price - cost priceSince to make minimum profit ; profit =0then;[tex]0= 2x-((\frac{-1}{250})x^2+2x+120)[/tex] or[tex]0 = 2x +\frac{1}{250}x^2-2x-120[/tex] Simplify:[tex]\frac{x^2}{250}- 120 =0[/tex]⇒ [tex]x^2= 30000[/tex]  or[tex]x =\sqrt{30000}[/tex]Simplify:x =173.205081orx = 174 (approx)Therefore, the minimum number of pens the company must sell to make a profit is, 174