Find the total cost of producing 5 widgets. Widget Wonders produces widgets. They have found that the cost, c(x), of making x widgets is a q

Find the total cost of producing 5 widgets. 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 $15.50 to produce 3 widgets, $23.50 to produce 7 widgets, and $56 to produce 12 widgets.

OK...so we have

a(7)^2 + b(7) + c =  23.50 →   49a + 7b + c = 23.50    (1)

a(3)^2 + b(3) + c =  15.50 →     9a  + 3b + c = 15.50         subtracting the second equation from the first, we have

40a + 4b = 8  → 10a + b = 2    (2)

Also

a(12)^2 + b(12) + c = 56  → 144a + 12b + c = 56   and subtracting (1) from this gives us

95a + 5b = 32.50

And using(2)  we have

95a + 5b = 32.50   (3)

10a + b  =    2.00        multiplying the second equation by -5  and adding  this to (3) ,we have

45a = 22.50      divide both sides by 45    and a = 1/2   and using (2) to find b, we have

10(1/2) + b = 2

5 + b = 2      b = -3

And we can use 9a  + 3b + c = 15.50 to find "c"

9(1/2) + 3(-3) + c = 15.50

9/2 - 9  + c = 15.50

-4.5 + c = 15.50      

c = 20

So our function is

c(x) = (1/2)x^2 - (3)x  + 20

And the cost to produce 5 widgets is = $17.50

Here's the graph of our function

Find the total cost of producing 6 widgets.

 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 $23 to produce 2 widgets, $103 to produce 4 widgets, and $631 to produce 10 widgets.