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Question: A spring bounces up and down according to the model d(t) = 4cos(30t) − 1, where d(t) is the displacement in cm from the rest position and t is time in seconds. What is the range?
The range of a function is the set of possible output values (in this case, the displacement d(t)). Here, the function d(t) = 4cos(30t) − 1 describes a cosine wave that varies vertically, and the 4 in front of the cosine function controls its amplitude, the peak deviation of the function from zero. Cosine's maximum value is 1 and its minimum value is -1. If we multiply this by 4, we find that the maximum and minimum values for the "4cos(30t)" component of the function are 4 and -4. The “-1” outside the function corresponds to a vertical shift of the graph - it moves the whole graph 1 unit downwards. Therefore, when we apply it, the amplitude value (from -4 to 4) is shifted down by 1 unit, changing the range to -5 to 3. So, the range of the displacement d(t) in cm for this spring is from -5 cm to 3 cm.
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