Widget wonders 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 $16 to produce 2 widgets, $18 to produce 4 widgets, and $48 to produce 10 widgets. find the total cost of producing 6 widgets.

Below is the complete solution explained step-by-step.

The answers are:

Function: c(x) = (1/2)x^2 - 2x + 18

Cost to produce 6 widgets: $24

Explanation:

The general form of a quadratic function is:

c(x) = Ax^2 + Bx + C

So, you need to find the values of A, B, C to know the quadratic function.

Those are three unknowns. Since, you have three pair of values, you can set a systmen of three equations with trhee unknown variables.

This is it:

1) x = 2 widgets,  c(x) = $ 16 => A(2)^2 + B(2) + C = 16

2) x = 4 widgets, c(x) = $18 => A(4^2) + B(4) + C = 18

3) x = 10 widgets, c(x) = $48 => A(10^2) + B(10) + C = 48

Developing the expressions you get:

1) 4A + 2B + C = 16
2) 16A + 4B + C = 18
3) 100A + 10B + C = 48

Now solve the system. I recommend to eliminate C to make two equations with A and B.

Equation 2 - Equation 1=> (3) 12A + 2B = 2

Equation 3 - Equation 2 => (4) 84A + 6B = 30

Multiply equation (3) times 3 and subtract from equation (4) =>

84A + 6B = 30
36A + 6B = 6

84A - 36A = 30 - 6
48A = 24
A = 24/48
A = 1/2

Replace A = 1/2 on 12A + 2B = 2

=> 12(1/2) + 2B = 2 => 6 + 2B = 2 => 2B = 2 - 6 =  -4

=> B = -4 / 2 = -2

Replace A and B on 4A + 2B + C = 16

=> 4(1/2) + 2(-2) + C = 16
=> C = 16 - 2 + 4 = 18

Therefore, the quatratic function is c(x) = (1/2)x^2 - 2x + 18.

c(x) = (1/2)x^2 - 2x + 18

To find the cost to produce 6 widgets, substitute x = 6:

=> (1/2) (6^2) - 2(6) + 18 = 36/2 - 12 + 18 = 18 - 12 + 18 = 24

$24.