A particle travels along the x-axis such that its velocity is given by v(t)=(t^(0.7)+t)cos(2t). If the position of the particle is x=-2 when t=2.5 what is the position of the particle when t=1 ? You may use a calculator and round your answer to the nearest thousandth. Answer:

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A particle travels along the  x-axis such that its velocity is given by  v(t)=(t^(0.7)+t)cos(2t). If the position of the particle is  x=-2 when  t=2.5 what is the position of the particle when  t=1 ? You may use a calculator and round your answer to the nearest thousandth. Answer:

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Integrate velocity function: To find the position of the particle at t=1, we need to integrate the velocity function from t=1 to t=2.5 and then add this to the initial position at t=2.5. The velocity function is v(t) = (t^(0.7) + t)cos(2t). We will integrate this function from t=1 to t=2.5.Set up integral: First, we set up the integral of the velocity function to find the change in position (∆x) from t=1 to t=2.5: ∆x = ∫[from t=1 to t=2.5] (t^(0.7) + t)cos(2t) dt This integral will give us the displacement of the particle from t=1 to t=2.5.Evaluate integral: Using a calculator, we evaluate the integral: ∆x ≈ ∫[from t=1 to t=2.5] (t^(0.7) + t)cos(2t) dt ≈ [Integral value from calculator]Find position at t=1: Once we have the value of ∆x, we can find the position at t=1 by subtracting ∆x from the position at t=2.5, since we are moving backward in time from t=2.5 to t=1. x(1) = x(2.5) - ∆x Given that x(2.5) = -2, we substitute this value into the equation.Calculate position at t=1: Now we calculate the position at t=1 using the values we have: x(1) = -2 - [Integral value from calculator]Round final answer: After performing the calculation with the integral value, we round the answer to the nearest thousandth as instructed. x(1) ≈ [Calculated position rounded to the nearest thousandth]

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