Answer:
Option B is correct
The 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]
or
1 = -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 =2
Therefore, 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 price
Since to make minimum profit ; profit =0
then;
[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 =173.205081
or
x = 174 (approx)
Therefore, the minimum number of pens the company must sell to make a profit is, 174
