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 8 widgets.

The answer is $34.

This is the system of three equations. Since it is quadratic function, the basic formula is
c(x) = ax² + bx + c,
where x is the number of widgets produced, and c(x) is cost to produce x number of widgets.
Thus, we need to calculate c(8). To do that, first we need to calculate a, b, and c from the quadratic formula using the system of 3 equations.

So, the equations are:
1. c(2) = 16= a(2)² + b(2) + c = 4a + 2b + c                 ⇒ 4a + 2b + c = 16
2. c(4) = 18= a(4)² + b(4) + c = 16a + 4b + c               ⇒ 16a + 4b + c = 18
3. c(10) = 48= a(10)² + b(10) + c = 100a + 10b + c     ⇒ 100a + 10b + c = 48

If we subtract equation 1 from equation 2, we have:
16a + 4b + c - 4a - 2b - c = 18 - 16             ⇒ 12a + 2b = 2
If we subtract equation 1 from equation 3, we have:
100a + 10b + c - 4a - 2b - c = 48 - 16         ⇒ 96a + 8b = 32


We have two equations now, and let's multiply first by 4 and the solve it:
12a + 2b = 2       /*4
96a + 8b = 32

48a + 8b = 8
96a + 8b = 32

We can now subtract these equations:
96a + 8b - 48a - 8b = 32 - 8
48a = 24         ⇒     a = 24/48 = 1/2

If we know a, we can calculate b from the equation:
12a + 2b = 2 
2b = 2 - 12a = 2 - 12 * 1/2 = 2 - 6 = -4
b = -4 ÷ 2 = -2

We have a and b. Let's calculate c:
4a + 2b + c = 16
c = 16 - 4a - 2b = 16 - 4 * 1/2 - 2 * (-2) = 16 - 2 + 4 = 18

Thus:
a = 1/2
b = -2
c = 18

It is easy to calculate c(8)
 c(8) = 1/2(8)² - 2(8) + 18 = 1/2 * 64 - 16 + 18 = 32 - 16 + 18 = 34

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