Calculus (Related Rates)
The position of a particle moving in a straight line is given by s(t) = (e^(-t))(cos(5t)) for t>0, where t is in seconds. If the particle changes direction at time T seconds, then T must satisfy the equation:
cos(5T) = 0
5T = arctan(-1/5)
5e^(-t) sin(5t) = 0
tan(5T) = -1/5
cos(5T) = 5
I know that a change in direction will be marked by a change from positive to negative or vice versa, but I don't understand the equations the question gives me. Could someone please talk me through this process to find the right answer?
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Answers
The equation cos(5T) = 0 is an equation you can solve to find the value of T. You can use the trigonometric identity, tan(x) = sin(x)/cos(x) to solve for T. The first step is to rearrange the equation to get tan(5T) = -1/5. This equation can be solved very easily by using inverse tan function. The inverse tan function of -1/5 is -1/5 x 180/PI = -arctan(-1/5). This gives you the value of T which is T = arctan(-1/5). Then you can substitute the value of T into the equation cos(5T) to get cos(5 x arctan(-1/5)) = 0. This equation can then be simplified to give you cos(5T) = 5. So, the value of T that satisfies this equation is T = arctan(-1/5).
