**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?**

**Answer:**

Let say cost c(x) depends as following –

C(x) = ax^2 + b x +c ------------ (1)

Where a, b, c are arbitrary constants and x is the number of widgets

Cost for 2 widgets = 23 so put x = 2 in equation 1

4a + 2b +c = 23 ---------- (2)

Cost for 4 widgets = 55 so put x = 4 in equation (1)

16 a +4b + c = 55 ------ (3)

Subtract equation (2) from equation (3)

12a + 2 b = 32 dividing both sides by 2

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

Given cost for 10 widgets = 247 so put x = 10 in equation (1)

100a +10b + c = 247 ----- (5)

Subtract equation 3 from equation 5

84a + 6b = 192 dividing both sides by 6

14a + b = 32 ------- (6)

Subtracting by equation 4 by equation 6

8a + 16

We get a =2

Now put a =2 in equation 4 we get

b = 4

Now put a and b values in equation 2

We get c = 7

By putting a =2, b = 4 and c = 7 in equation (1) we are getting the cost function c(x)

C(x) = 2x^2 + 4x + 7 ------ (7)

Now we have to find the cost of 8 widgets so put = 8 in equation 7

C(8) = 2*8*8 + 4*8 + 7

C(8) = 167

So production cost for 8 widgets is $167