Final answer:
To determine when water remains in the tank, we need to find when the cubic function f(x) is greater than zero. Since cubic functions cross the x-axis at their roots, the intervals where f(x) is positive indicate when there is water in the tank. The correct time period, given the function remains positive and includes the endpoints where the tank becomes empty, is (−[infinity], −8] ∪ [−5, 3].
Step-by-step explanation:
The function f(x) = −x³ − 10x² − x + 120 models the volume of water in a tank over time, where x is the number of minutes since the leak began. To find the time period during which there is water in the tank, we look for the range of x values for which f(x) is positive, since a negative volume wouldn't make sense physically. This implies we need to solve for when f(x) > 0.
We can find the roots of the polynomial, either by graphing the function or by analytical methods such as factoring or using the Rational Root Theorem, but the exact roots are not provided in the question or the given information. However, the answer options suggest there might be intervals before and after x lies between −∞ and −8, as well as between −5 and 3.
Typically, the behavior of a cubic function will result in it crossing the x-axis at its roots and changing signs, which means that the function will only be positive in certain intervals. Based on the usual trends of a cubic function's graph and the provided answer choices, we can infer that the correct answer would likely indicate the intervals during which f(x) is above the x-axis.
The answer choice that suggests f(x) is positive between two intervals, and includes the endpoints of the intervals, is the correct one, particularly if the endpoints represent the roots where the volume becomes zero. Thus, the correct answer should be the one that includes the intervals where the function is above zero and accounts for the endpoints being part of the positive volume. Of the options provided, the one that accounts for the intervals and includes the endpoints is Choice B: (−[infinity], −8] ∪ [−5, 3].