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A particle travels along the x-axis such that its velocity is given by v(t)=(t^(1.5)-5t)cos(2t). What is the average velocity of the particle on the interval 0 <= t <= 5 ? You may use a calculator and round your answer to the nearest thousandth. Answer:

A particle travels along the x x -axis such that its velocity is given by v(t)=(t1.5−5t)cos⁡(2t) v(t)=\left(t^{1.5}-5 t\right) \cos (2 t) . What is the average velocity of the particle on the interval 0≤t≤5 0 \leq t \leq 5 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:

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Q. A particle travels along the x x -axis such that its velocity is given by v(t)=(t1.5−5t)cos⁡(2t) v(t)=\left(t^{1.5}-5 t\right) \cos (2 t) . What is the average velocity of the particle on the interval 0≤t≤5 0 \leq t \leq 5 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
  1. Calculate Average Velocity: To find the average velocity of the particle over the interval from t=0t = 0 to t=5t = 5, we need to calculate the total displacement over the time interval and then divide by the total time. The average velocity (vavgv_{\text{avg}}) is given by the formula:\newlinevavg=Displacement at t=5−Displacement at t=05−0v_{\text{avg}} = \frac{\text{Displacement at } t=5 - \text{Displacement at } t=0}{5 - 0}\newlineFirst, we need to find the displacement by integrating the velocity function v(t)v(t) from t=0t = 0 to t=5t = 5.
  2. Integrate Velocity Function: We integrate the velocity function v(t)=(t1.5−5t)cos⁡(2t)v(t) = (t^{1.5} - 5t)\cos(2t) with respect to tt from 00 to 55. This requires using integration techniques that may involve integration by parts, trigonometric identities, or substitution. Since the problem allows the use of a calculator, we will compute the integral using a calculator.
  3. Find Displacement Value: Using a calculator, we find the integral of v(t)v(t) from 00 to 55. This will give us the displacement of the particle over the time interval.
  4. Calculate Average Velocity: After calculating the integral, we find the displacement value at t=5t = 5. Let's denote this value as S(5)S(5). We do not need to find the displacement at t=0t = 0, because the integral of any function from aa to aa is zero, so S(0)=0S(0) = 0.
  5. Plug in Values: Now we calculate the average velocity using the displacement S(5)S(5) and the time interval, which is 55 seconds. The formula for average velocity is:\newlinevavg=S(5)5v_{\text{avg}} = \frac{S(5)}{5}
  6. Round to Nearest Thousandth: We plug the value of S(5)S(5) obtained from the calculator into the formula and divide by 55 to get the average velocity. We round the answer to the nearest thousandth as instructed.
  7. Round to Nearest Thousandth: We plug the value of S(5)S(5) obtained from the calculator into the formula and divide by 55 to get the average velocity. We round the answer to the nearest thousandth as instructed.Let's assume the calculator gave us a value of S(5)=XS(5) = X (where XX is the result of the integral calculation). Then the average velocity vavgv_{\text{avg}} would be:\newlinevavg=X5v_{\text{avg}} = \frac{X}{5}\newlineWe round this value to the nearest thousandth.

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