A particle travels along the $x$ -axis such that its velocity is given by $v(t)=\left(t^{1.6}+3\right) \cos (3 t)$ . What is the average velocity of the particle on the interval $0 \leq t \leq 6$ ? You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

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Q. A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{1.6}+3\right) \cos (3 t)

. What is the average velocity of the particle on the interval

0 \leq t \leq 6

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Answer:

Set up integral: To find the average velocity of the particle over the interval from $t = 0$ to $t = 6$ , we need to integrate the velocity function $v(t)$ over this interval and then divide by the length of the interval. $\newline$ The average velocity formula is given by: $\newline$ Average velocity = $\frac{1}{(b-a)} \times \int_{a}^{b} v(t) \, dt$ $\newline$ Here, $a = 0$ , $b = 6$ , and $v(t) = (t^{1.6}+3)\cos(3t)$ .
Evaluate integral: First, we set up the integral for the average velocity: $\newline$ Average velocity = $(1/(6-0)) \times \int_{0}^{6} (t^{1.6}+3)\cos(3t) \, dt$
Divide by interval: Next, we use a calculator to evaluate the integral. Since the integral involves both a power of $t$ and a trigonometric function, it's not straightforward to integrate by hand. We rely on numerical methods or a calculator's built-in integration function.
Round result: After evaluating the integral from $0$ to $6$ of $(t^{1.6}+3)\cos(3t)$ dt using a calculator, we get a numerical value. Let's assume this value is $I$ (since we don't have an actual calculator to perform the computation).
Round result: After evaluating the integral from $0$ to $6$ of $(t^{1.6}+3)\cos(3t) \, dt$ using a calculator, we get a numerical value. Let's assume this value is $I$ (since we don't have an actual calculator to perform the computation).Now, we divide the value of the integral $I$ by the length of the interval, which is $6 - 0 = 6$ , to find the average velocity. $\newline$ Average velocity = $I / 6$
Round result: After evaluating the integral from $0$ to $6$ of $(t^{1.6}+3)\cos(3t) \, dt$ using a calculator, we get a numerical value. Let's assume this value is $I$ (since we don't have an actual calculator to perform the computation).Now, we divide the value of the integral $I$ by the length of the interval, which is $6 - 0 = 6$ , to find the average velocity. $\newline$ Average velocity = $I / 6$ We round the result to the nearest thousandth as instructed. $\newline$ Assuming the calculator gave us the value of $I$ , we would then divide it by $6$ and round it. Let's say the calculator gives us $I = 15.678$ (this is a hypothetical value for the sake of demonstration). $\newline$ Average velocity = $6$ $0$ $\newline$ Average velocity $6$ $1$ (rounded to the nearest thousandth)