Final answer:
In high school mathematics, solving a piecewise function or a quadratic equation involves using different methods. For piecewise functions, you evaluate the equation based on the intervals of the variable. For quadratic equations, the quadratic formula (-b ± √(b² - 4ac) / (2a)) is typically used to find the solutions.
Step-by-step explanation:
The question involves solving a piecewise function for a variable t. Initially, we have two equations describing the behavior of U(t) in different time intervals:
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- U(t) = -378t + 675 for 0 ≤ t ≤ 2
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- U(t) = 358t - 842 for 2 < t < 6
To find the values of t, you will need to solve these equations based on the given conditions. The mentioned quadratic equation t² + 10t - 200 = 0, which can be solved using the quadratic formula, is different from the piecewise function described but demonstrates the method one would use to find t when dealing with such equations.
The correct application of the quadratic formula is vital for finding solutions to quadratic equations. The formula is: -b ± √(b² - 4ac) / (2a). This will generally yield two solutions, which in the context given, are t = 3.96 and t = -1.03. The negative time value usually signifies that the event occurred before the start of observation, which in many physical contexts is disregarded, leaving you with the positive value, t = 3.96 seconds.