Company ABC produces widgets. They have found that the cost, c (x), of making x widgets is a quadratic function in terms of x. The company also discovered

Company ABC produces widgets. They have found that the cost,  c (x), of making x widgets is a quadratic function in terms of x. The company also discovered that it costs $23 to produce 2 widgets, $55 to produce 4 widgets, and $247 to produce 10 widgets. What is the total cost of producing 8 widgets?

We can solve this with the following system

a(2)^2  + b(2) + c = 23

a(4)^2 + b(4) + c = 55

a(10)^2 + b(10) + c  = 247      simplifying, we have

4a + 2b + c  = 23            (1)

16a + 4b + c = 55           (2)

100a + 10b + c   = 247     (3)

Subtract (1) from (2)    and  (2) from (3)    ...and we get the following system

12a + 2b  =   32

84a + 6b  =  192       these simplify to

6a + b = 16    →    b =  16 - 6a    (4)

28a + 2b  = 64        (5)

Substitute   (4) into (5)

28a + 2[16 - 6a] = 64   simplify

28a + 32 - 12a  = 64

16a + 32 = 64    subtract 32 from both sides

16a  = 32      divide both sides by 16

a = 2

And using (4)  .....

  b =  16 - 6(2)  = 16 - 12  = 4

And using (1) ......

4(2) + 2(4) + c = 23

8 + 8 + c = 23

16 + c = 23

So    c  = 7

And our cost function is :

c(x)  = 2x^2 + 4x + 7       and the cost to produce 8 widgets is

c(8)  = 2(8)^2 + 4(8) + 7   =   2*64 + 32 + 7  =  128 + 39  = $ 167