Final answer:
To find the number of flags manufactured when the cost is $147, the quadratic cost function c = x^2 - 12x + 39 is set to 147. The quadratic formula is applied to solve x^2 - 12x - 108 = 0, giving two solutions, 18 and -6, with only 18 being a viable answer. So 18 flags wiil be produced.
Step-by-step explanation:
The student asked to find the number of flags manufactured if the cost is $147, based on the given quadratic cost function c = x^2 - 12x + 39. To find the number of flags (x), we set the cost function equal to $147 and solve for x:
c = x^2 - 12x + 39
147 = x^2 - 12x + 39
x^2 - 12x + 39 - 147 = 0
x^2 - 12x - 108 = 0
Now we use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients from the quadratic equation ax^2 + bx + c = 0. Substituting the appropriate values into the formula gives us two possible solutions for x:
x = (12 ± √((12)^2 - 4(1)(-108))) / (2(1))
x = (12 ± √(144 + 432)) / 2
x = (12 ± √576) / 2
x = (12 ± 24) / 2
x = 18 or x = -6 (which is not a viable solution since the number of flags cannot be negative)
The number of flags manufactured, when the cost is $147, is 18 flags.