Final answer:
The domain of the function C(t) = 6t / (12t^2 + 3) is all real numbers because the denominator 12t^2 + 3 never equals zero for any real number t.
Step-by-step explanation:
To solve for the domain of the given function C(t) = 6t / (12t^2 + 3), we need to identify all values of t for which the function is defined. The domain of a function includes all the input values (here, t) for which the function gives a real number as an output.
In the case of rational functions like this one, the domain is all real numbers except where the denominator equals zero. Therefore, we need to ensure that the denominator 12t^2 + 3 ≠ 0.
Since the denominator is a quadratic expression that is always positive (as 12t^2 will always be non-negative and 3 is positive), the equation 12t^2 + 3 = 0 has no real solutions.
Thus, there are no restrictions on t, and the domain of C(t) is all real numbers.