We can solve this with the following system
a(2)^2 + b(2) + c = 23
a(4)^2 + b(4) + c = 55
a(10)^2 + b(10) + c = 247 simplifying, we have
4a + 2b + c = 23 (1)
16a + 4b + c = 55 (2)
100a + 10b + c = 247 (3)
Subtract (1) from (2) and (2) from (3) ...and we get the following system
12a + 2b = 32
84a + 6b = 192 these simplify to
6a + b = 16 → b = 16 - 6a (4)
28a + 2b = 64 (5)
Substitute (4) into (5)
28a + 2[16 - 6a] = 64 simplify
28a + 32 - 12a = 64
16a + 32 = 64 subtract 32 from both sides
16a = 32 divide both sides by 16
a = 2
And using (4) .....
b = 16 - 6(2) = 16 - 12 = 4
And using (1) ......
4(2) + 2(4) + c = 23
8 + 8 + c = 23
16 + c = 23
So c = 7
And our cost function is :
c(x) = 2x^2 + 4x + 7 and the cost to produce 8 widgets is
c(8) = 2(8)^2 + 4(8) + 7 = 2*64 + 32 + 7 = 128 + 39 = $ 167
