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

Well the table function I got was 2 x^2+4 x+7

So I got 8 widgets = $167