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A particle travels along the x-axis such that its velocity is given by v(t)=t^(2.1)sin(2t+5). Find all times when the speed of the particle is equal to 1 on the interval 0 <= t <= 3. You may use a calculator and round your answer to the nearest thousandth. Answer: t=

A particle travels along the x x -axis such that its velocity is given by v(t)=t2.1sin⁡(2t+5) v(t)=t^{2.1} \sin (2 t+5) . Find all times when the speed of the particle is equal to 11 on the interval 0≤t≤3 0 \leq t \leq 3 . You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer: t= t=

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Q. A particle travels along the x x -axis such that its velocity is given by v(t)=t2.1sin⁡(2t+5) v(t)=t^{2.1} \sin (2 t+5) . Find all times when the speed of the particle is equal to 11 on the interval 0≤t≤3 0 \leq t \leq 3 . You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer: t= t=
  1. Understand the Problem: Understand the problem.\newlineWe need to find the times when the speed of the particle is equal to 11. The speed is the absolute value of the velocity, so we are looking for when ∣v(t)∣=1|v(t)| = 1.
  2. Set Up the Equation: Set up the equation.\newlineWe have the velocity function v(t)=t2.1sin⁡(2t+5)v(t) = t^{2.1}\sin(2t+5). To find when the speed is 11, we set up the equation ∣t2.1sin⁡(2t+5)∣=1|t^{2.1}\sin(2t+5)| = 1.
  3. Solve the Equation: Solve the equation using a calculator.\newlineSince the equation involves both a power function and a trigonometric function, it's not easily solvable by algebraic means. We will use a calculator to find the values of tt in the interval [0,3][0, 3] that satisfy the equation.
  4. Check Calculator Settings: Check the calculator's settings.\newlineBefore using the calculator, ensure it is set to the correct mode (radians or degrees) based on the function given. Since the function involves sin⁡(2t+5)\sin(2t+5), and there is no indication that we should use degrees, we will assume the calculator should be in radian mode.
  5. Use the Calculator: Use the calculator to find the values of tt. Input the function ∣t2.1sin⁡(2t+5)∣|t^{2.1}\sin(2t+5)| into the calculator and use a graphing or solver feature to find the points where the function crosses the line y=1y = 1 within the interval [0,3][0, 3]. Note down the values of tt that satisfy the equation, rounding to the nearest thousandth as instructed.
  6. Verify the Solutions: Verify the solutions.\newlineCheck each value of tt obtained from the calculator to ensure that when plugged back into the function ∣t2.1sin⁡(2t+5)∣|t^{2.1}\sin(2t+5)|, it indeed gives a value of 11. This is to confirm that there were no input errors when using the calculator.
  7. List Valid Times: List all valid times.\newlineAfter verifying, list all the times that satisfy the equation within the given interval. These are the times when the speed of the particle is equal to 11.

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