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.
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
subtract equation 1 from equation 2:
16a + 4b + c - 4a - 2b - c = 18 - 16 = 12a + 2b = 2
subtract equation 1 from equation 3:
100a + 10b + c - 4a - 2b - c = 48 - 16 = 96a + 8b = 32
We have two equations now, multiply the first by 4 ( to equal out b):
12a + 2b = 2 = 48a +8b =8
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
so a = ½ , b = -2, c = 18
now calculate c(6)
c(6) = 1/2(6)² - 2(6) + 18 = 1/2 * 36 - 12 + 18 = 18 - 12 + 18 = 24
it costs $24 to produce 6 widgets