Question

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 1

Its B I got it right on edge

Step-by-step explanation:

Answer 2

Option B is correct

The minimum number of pens the company must sell to make a profit is, 174.

Step-by-step 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

`y=ax^2+bx+c` ......[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:

`x =(-b)/(2a)` ......[2]

Substitute the value of x = 250 in [2];

`250 = (-b)/(2a)` or

`500a = -b` ......[3]

Substitute the value of x=250, y =370, c =120 and b = -500 a in [1];

`370=a(250)^2+(-500a)(250)+120` or

`250 = a(250)^2-(500a)(250)` or

`1 = 250a -500 a`

or

1 = -250 a

⇒
`a= (-1)/(250)`

We put the value of a in [3]

So,

b =-500 a=
`-500 \cdot (-1)/(250)`

Simplify:

b =2

Therefore, the cost price of the pens is:
`y = ((-1)/(250))x^2+2x+120`

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;

`0= 2x-(((-1)/(250))x^2+2x+120)` or

`0 = 2x +(1)/(250)x^2-2x-120`

Simplify:

`(x^2)/(250)- 120 =0`

⇒
`x^2= 30000` or

`x =√(30000)`

Simplify:

x =173.205081

or

x = 174 (approx)

Therefore, the minimum number of pens the company must sell to make a profit is, 174