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
......[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:
......[2]
Substitute the value of x = 250 in [2];
or
......[3]
Substitute the value of x=250, y =370, c =120 and b = -500 a in [1];
or
or
or
1 = -250 a
⇒![a= \frac{-1}{250}](https://tex.z-dn.net/?f=a%3D%20%5Cfrac%7B-1%7D%7B250%7D)
We put the value of a in [3]
So,
b =-500 a= ![-500 \cdot \frac{-1}{250}](https://tex.z-dn.net/?f=-500%20%5Ccdot%20%5Cfrac%7B-1%7D%7B250%7D)
Simplify:
b =2
Therefore, the cost price of the pens is: ![y = (\frac{-1}{250})x^2+2x+120](https://tex.z-dn.net/?f=y%20%3D%20%28%5Cfrac%7B-1%7D%7B250%7D%29x%5E2%2B2x%2B120)
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;
or
Simplify:
![\frac{x^2}{250}- 120 =0](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B250%7D-%20120%20%3D0)
⇒
or
![x =\sqrt{30000}](https://tex.z-dn.net/?f=x%20%3D%5Csqrt%7B30000%7D)
Simplify:
x =173.205081
or
x = 174 (approx)
Therefore, the minimum number of pens the company must sell to make a profit is, 174