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

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

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

$x =\frac{-b}{2a}$                 ......[2]

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

$250 = \frac{-b}{2a}$ 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

⇒$a= \frac{-1}{250}$

We put the value of a in [3]

So,

b =-500 a= $-500 \cdot \frac{-1}{250}$

Simplify:

b =2

Therefore, the cost  price of the pens is:  $y = (\frac{-1}{250})x^2+2x+120$

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;

$0= 2x-((\frac{-1}{250})x^2+2x+120)$ or

$0 = 2x +\frac{1}{250}x^2-2x-120$

Simplify:

$\frac{x^2}{250}- 120 =0$

⇒ $x^2= 30000$  or

$x =\sqrt{30000}$

Simplify:

x =173.205081

or

x = 174 (approx)

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

Answer 2

Imma guess, b .............