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?

Answer 1

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

......[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

⇒

We put the value of a in [3]

So,

b =-500 a=

Simplify:

b =2

Therefore, the cost price of the pens is:

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:

⇒ or

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

y = ax^2 + bx + c

Since y-intercept is 120 this means for x = 0, y = 120 that is:

a*0^2 + b*0 + c = 120

c = 120

The vertex is given by (-b/2a, (c - b^2)/4a) this means:

-b/2a = 250

b = -500a

Since the line goes trough (250, 370), that is for

x = 250,

y = 370:

a*250^2 + (-500a)*250 + 120

= 370

a = -0.004

b = -500*(-0.004)

b = 2

The equation is:

y = -0.004x^2 + 2x + 120

The company makes profit when y > 2x that is:

-0.004x^2 + 2x + 120 > 2x

-0.004x^2 > - 120

x^2 < 120/0.004

x^2 < 30000

x < 173.2

Thus the minimum number of pens the company must sell is 1.