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A particle travels along the
x-axis such that its velocity is given by
v(t)=t^(2)sin(2t). Find all times when the speed of the particle is equal to 3 on the interval
0 <= t <= 4. You may use a calculator and round your answer to the nearest thousandth.
Answer:
t=

A particle travels along the $x$ -axis such that its velocity is given by $v(t)=t^{2} \sin (2 t)$ . Find all times when the speed of the particle is equal to $3$ on the interval $0 \leq t \leq 4$ . You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer: $t=$

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Write linear functions: word problems

Full solution

Q. A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{2} \sin (2 t)

. Find all times when the speed of the particle is equal to

3

on the interval

0 \leq t \leq 4

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

\newline

Answer:

t=

Understand Speed vs Velocity: Understand the relationship between speed and velocity. Speed is the absolute value of velocity. Therefore, to find when the speed is equal to $3$ , we need to solve the equation $|v(t)| = 3$ .
Set up Equation for t: Set up the equation to solve for $t$ . We have $v(t) = t^2 \sin(2t)$ , so we need to solve $|t^2 \sin(2t)| = 3$ .
Solve Equation Using Calculator: Solve the equation for $t$ using a calculator. $\newline$ Since this equation is not easily solvable by algebraic methods, we will use a calculator to find the values of $t$ in the interval $[0, 4]$ that satisfy the equation. We are looking for points where the graph of $y = t^2 \sin(2t)$ crosses $y = 3$ and $y = -3$ .
Check Calculator's Solutions: Check the calculator's solutions. $\newline$ After using the calculator, we find that the graph of $y = t^2 \cdot \sin(2t)$ crosses $y = 3$ and $y = -3$ at certain points within the interval $[0, 4]$ . We round these solutions to the nearest thousandth.
Verify Solutions in Interval: Verify the solutions are within the given interval. $\newline$ We need to ensure that the solutions obtained from the calculator are within the interval $0 \leq t \leq 4$ .
List Valid Times: List all valid times $t$ . Assuming the calculator gave us valid times within the interval, we list these times as the final answer.

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