Final answer:
The value of t when D(t) = 1,200 needs the specific decay formula or data on D(t) to provide an exact answer. It likely represents time elapsed and the context hints at an exponential decay process. Without further context, we can reference similar problems involving decay or growth, but a precise solution cannot be given.
Step-by-step explanation:
The value of t when D(t) = 1,200 can be found by solving the equation given by the context of the problem—which appears to relate to exponential decay, possibly in a physics or chemistry context. It implies that t is the time elapsed since the start of the process, perhaps the decay of a radioactive substance or the discharge of a capacitor in electronics. Given that t is supposed to be larger than the half-life because the activity has dropped to less than half, we can infer that the function D(t) represents some kind of decaying quantity over time. To find t, you would typically use the decay formula or refer to provided data or a graph.
If we refer to an example given where the decay process is described, the value of t can be obtained by setting up the equation and solving for t. This could be done using the natural logarithm if the process is exponential. Additionally, the information provided about the rise of current in a circuit over time using a characteristic time τ and an exponential relationship could be an analogy to find the behavior of D(t) over time. However, without the specific decay formula or data on D(t), we cannot provide the numerical answer.
If the equation D(t) takes the form of a quadratic function as suggested by the details given—t² + 10t - 2000—you would solve for t using the quadratic formula:
−b ± √(b² - 4ac) / (2a)
Substituting the coefficients a = 1, b = 10, and c = -2000, you can find the value for t.